/-
 - Created in 2024 by Gaëtan Serré
-/

/-
 - https://github.com/gaetanserre/SBS-Proofs
-/

import Mathlib

import SBSProofs.Utils

open Classical MeasureTheory

local macro_rules | `($x ^ $y) => `(HPow.hPow $x $y)

set_option trace.Meta.Tactic.simp.rewrite true
set_option maxHeartbeats 600000

variable {d : ℕ} {Ω : Set (Vector ℝ d)} [MeasureSpace Ω]

def L2 (μ : Measure Ω) [IsFiniteMeasure μ] := {f : Ω → ℝ | Memℒp f 2 μ}

def eigen := {e : ℕ → ℝ // ∀ i, 0 <= e i}

def f_repr (v : eigen) (e : ℕ → Ω → ℝ) (f : Ω → ℝ) (a : ℕ → ℝ) := (f = λ x ↦ (∑' i, (v.1 i) * (a i) * (e i x))) ∧ (∀ x, Summable (λ i ↦ (v.1 i) * (a i) * (e i x)))

/-
  We define a set of functions that depends on a finite measure μ. Each function is representable by a infinite sum.
-/

def H (v : eigen) (e : ℕ → Ω → ℝ) (μ : Measure Ω) [IsFiniteMeasure μ] := {f | f ∈ L2 μ ∧ ∃ (a : ℕ → ℝ), (f_repr v e f a) ∧ Summable (λ i ↦ (v.1 i) * (a i)^2)}

def set_repr {v : eigen} {e : ℕ → Ω → ℝ} {μ : Measure Ω} [IsFiniteMeasure μ] (f : H v e μ) := {a : ℕ → ℝ | (f_repr v e f.1 a) ∧ (Summable (λ i ↦ (v.1 i) * (a i)^2))}

lemma set_repr_ne {v : eigen} {e : ℕ → Ω → ℝ} {μ : Measure Ω} [IsFiniteMeasure μ] (f : H v e μ) : (set_repr f).Nonempty := by
Goals accomplished! 🐙
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f: ↑(H v e μ)
Set.Nonempty (set_repr f)rcases f.2 with ⟨_, ⟨a, ha⟩⟩
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f: ↑(H v e μ)
left✝: ↑f ∈ L2 μ
a: ℕ → ℝ
ha: f_repr v e (↑f) a ∧ Summable fun i => ↑v i * a i ^ 2
intro.intro
Set.Nonempty (set_repr f)
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f: ↑(H v e μ)
Set.Nonempty (set_repr f)use a
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f: ↑(H v e μ)
left✝: ↑f ∈ L2 μ
a: ℕ → ℝ
ha: f_repr v e (↑f) a ∧ Summable fun i => ↑v i * a i ^ 2
h
a ∈ set_repr f
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f: ↑(H v e μ)
Set.Nonempty (set_repr f)exact ha
Goals accomplished! 🐙

/-
  We assume that the the representative of each function in H is unique (property of v and e).
-/

axiom set_repr_unique {v : eigen} {e : ℕ → Ω → ℝ} {μ : Measure Ω} [IsFiniteMeasure μ] {f : H v e μ} {a b : ℕ → ℝ} (ha : a ∈ set_repr f) (hb : b ∈ set_repr f) : a = b

lemma unique_choice {v : eigen} {e : ℕ → Ω → ℝ} {μ : Measure Ω} [IsFiniteMeasure μ] {f : H v e μ} {a : ℕ → ℝ} (h : a ∈ set_repr f) : (set_repr_ne f).some = a := set_repr_unique (set_repr_ne f).some_mem h

/-
  We assume that the product of two representative is summable.
-/

axiom product_summable {v : eigen} {e : ℕ → Ω → ℝ} {μ : Measure Ω} [IsFiniteMeasure μ] (f g : H v e μ) : Summable (λ i ↦ (v.1 i) * ((set_repr_ne f).some i) * ((set_repr_ne g).some i))


/-
  We define the multiplication between a real number and a function in H as the pointwise product. We show that the result lies in H.
-/

namespace Ring

variable {v : eigen} {e : ℕ → Ω → ℝ} {μ : Measure Ω} [IsFiniteMeasure μ] (f g : H v e μ)

lemma mul_repr (a : ℝ) : f_repr v e (λ x ↦ a * f.1 x) (λ i ↦ a * (set_repr_ne f).some i) := by
Goals accomplished! 🐙
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
f_repr v e (fun x => a * ↑f x) fun i => a * Set.Nonempty.some ⋯ ilet g := λ x ↦ a * f.1 x
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
f_repr v e (fun x => a * ↑f x) fun i => a * Set.Nonempty.some ⋯ i
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
f_repr v e (fun x => a * ↑f x) fun i => a * Set.Nonempty.some ⋯ ilet h := (set_repr_ne f).some
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
f_repr v e (fun x => a * ↑f x) fun i => a * Set.Nonempty.some ⋯ i
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
f_repr v e (fun x => a * ↑f x) fun i => a * Set.Nonempty.some ⋯ ilet g_h := λ i ↦ a * h i
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
f_repr v e (fun x => a * ↑f x) fun i => a * Set.Nonempty.some ⋯ i
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
f_repr v e (fun x => a * ↑f x) fun i => a * Set.Nonempty.some ⋯ ihave f_repr : h ∈ set_repr f := (set_repr_ne f).some_mem
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
f_repr: h ∈ set_repr f
_root_.f_repr v e (fun x => a * ↑f x) fun i => a * Set.Nonempty.some ⋯ i
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
f_repr v e (fun x => a * ↑f x) fun i => a * Set.Nonempty.some ⋯ iconstructor
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
f_repr: h ∈ set_repr f
left
(fun x => a * ↑f x) = fun x => ∑' (i : ℕ), ↑v i * (fun i => a * Set.Nonempty.some ⋯ i) i * e i x
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
f_repr: h ∈ set_repr f
right
∀ (x : ↑Ω), Summable fun i => ↑v i * (fun i => a * Set.Nonempty.some ⋯ i) i * e i x
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
f_repr v e (fun x => a * ↑f x) fun i => a * Set.Nonempty.some ⋯ i·
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
f_repr: h ∈ set_repr f
left
(fun x => a * ↑f x) = fun x => ∑' (i : ℕ), ↑v i * (fun i => a * Set.Nonempty.some ⋯ i) i * e i x 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
f_repr: h ∈ set_repr f
left
(fun x => a * ↑f x) = fun x => ∑' (i : ℕ), ↑v i * (fun i => a * Set.Nonempty.some ⋯ i) i * e i x
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
f_repr: h ∈ set_repr f
right
∀ (x : ↑Ω), Summable fun i => ↑v i * (fun i => a * Set.Nonempty.some ⋯ i) i * e i xext x
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
f_repr: h ∈ set_repr f
x: ↑Ω
left.h
a * ↑f x = ∑' (i : ℕ), ↑v i * (fun i => a * Set.Nonempty.some ⋯ i) i * e i x
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
f_repr: h ∈ set_repr f
left
(fun x => a * ↑f x) = fun x => ∑' (i : ℕ), ↑v i * (fun i => a * Set.Nonempty.some ⋯ i) i * e i xrcases f_repr with ⟨⟨f_repr, _⟩, _⟩
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
x: ↑Ω
right✝¹: Summable fun i => ↑v i * h i ^ 2
f_repr: ↑f = fun x => ∑' (i : ℕ), ↑v i * h i * e i x
right✝: ∀ (x : ↑Ω), Summable fun i => ↑v i * h i * e i x
left.h.intro.intro
a * ↑f x = ∑' (i : ℕ), ↑v i * (fun i => a * Set.Nonempty.some ⋯ i) i * e i x
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
f_repr: h ∈ set_repr f
left
(fun x => a * ↑f x) = fun x => ∑' (i : ℕ), ↑v i * (fun i => a * Set.Nonempty.some ⋯ i) i * e i xhave g_eq_tsum : g = λ x ↦ ∑' i, v.1 i * a * h i * e i x := 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
x: ↑Ω
right✝¹: Summable fun i => ↑v i * h i ^ 2
f_repr: ↑f = fun x => ∑' (i : ℕ), ↑v i * h i * e i x
right✝: ∀ (x : ↑Ω), Summable fun i => ↑v i * h i * e i x
left.h.intro.intro
a * ↑f x = ∑' (i : ℕ), ↑v i * (fun i => a * Set.Nonempty.some ⋯ i) i * e i xby
Goals accomplished! 🐙 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
x: ↑Ω
right✝¹: Summable fun i => ↑v i * h i ^ 2
f_repr: ↑f = fun x => ∑' (i : ℕ), ↑v i * h i * e i x
right✝: ∀ (x : ↑Ω), Summable fun i => ↑v i * h i * e i x
g = fun x => ∑' (i : ℕ), ↑v i * a * h i * e i x{
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
x: ↑Ω
right✝¹: Summable fun i => ↑v i * h i ^ 2
f_repr: ↑f = fun x => ∑' (i : ℕ), ↑v i * h i * e i x
right✝: ∀ (x : ↑Ω), Summable fun i => ↑v i * h i * e i x
g = fun x => ∑' (i : ℕ), ↑v i * a * h i * e i x
      
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
x: ↑Ω
right✝¹: Summable fun i => ↑v i * h i ^ 2
f_repr: ↑f = fun x => ∑' (i : ℕ), ↑v i * h i * e i x
right✝: ∀ (x : ↑Ω), Summable fun i => ↑v i * h i * e i x
g = fun x => ∑' (i : ℕ), ↑v i * a * h i * e i xext x
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
x✝: ↑Ω
right✝¹: Summable fun i => ↑v i * h i ^ 2
f_repr: ↑f = fun x => ∑' (i : ℕ), ↑v i * h i * e i x
right✝: ∀ (x : ↑Ω), Summable fun i => ↑v i * h i * e i x
x: ↑Ω
h
g x = ∑' (i : ℕ), ↑v i * a * h i * e i x
      
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
x: ↑Ω
right✝¹: Summable fun i => ↑v i * h i ^ 2
f_repr: ↑f = fun x => ∑' (i : ℕ), ↑v i * h i * e i x
right✝: ∀ (x : ↑Ω), Summable fun i => ↑v i * h i * e i x
g = fun x => ∑' (i : ℕ), ↑v i * a * h i * e i xhave comm : ∀ i, v.1 i * a * h i * e i x = a * v.1 i * h i * e i x := λ i ↦ 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
x✝: ↑Ω
right✝¹: Summable fun i => ↑v i * h i ^ 2
f_repr: ↑f = fun x => ∑' (i : ℕ), ↑v i * h i * e i x
right✝: ∀ (x : ↑Ω), Summable fun i => ↑v i * h i * e i x
x: ↑Ω
h
g x = ∑' (i : ℕ), ↑v i * a * h i * e i xby
Goals accomplished! 🐙 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
x✝: ↑Ω
right✝¹: Summable fun i => ↑v i * h i ^ 2
f_repr: ↑f = fun x => ∑' (i : ℕ), ↑v i * h i * e i x
right✝: ∀ (x : ↑Ω), Summable fun i => ↑v i * h i * e i x
x: ↑Ω
i: ℕ
↑v i * a * h i * e i x = a * ↑v i * h i * e i xring
Goals accomplished! 🐙
      
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
x: ↑Ω
right✝¹: Summable fun i => ↑v i * h i ^ 2
f_repr: ↑f = fun x => ∑' (i : ℕ), ↑v i * h i * e i x
right✝: ∀ (x : ↑Ω), Summable fun i => ↑v i * h i * e i x
g = fun x => ∑' (i : ℕ), ↑v i * a * h i * e i xsimp_rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
x✝: ↑Ω
right✝¹: Summable fun i => ↑v i * h i ^ 2
f_repr: ↑f = fun x => ∑' (i : ℕ), ↑v i * h i * e i x
right✝: ∀ (x : ↑Ω), Summable fun i => ↑v i * h i * e i x
x: ↑Ω
comm: ∀ (i : ℕ), ↑v i * a * h i * e i x = a * ↑v i * h i * e i x
h
g x = ∑' (i : ℕ), ↑v i * a * h i * e i xcomm
[Meta.Tactic.simp.rewrite] comm:1000, ↑v i * a * h i * e i x ==> a * ↑v i * h i * e i x
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
x✝: ↑Ω
right✝¹: Summable fun i => ↑v i * h i ^ 2
f_repr: ↑f = fun x => ∑' (i : ℕ), ↑v i * h i * e i x
right✝: ∀ (x : ↑Ω), Summable fun i => ↑v i * h i * e i x
x: ↑Ω
comm: ∀ (i : ℕ), ↑v i * a * h i * e i x = a * ↑v i * h i * e i x
h
g x = ∑' (i : ℕ), a * ↑v i * h i * e i x]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
x✝: ↑Ω
right✝¹: Summable fun i => ↑v i * h i ^ 2
f_repr: ↑f = fun x => ∑' (i : ℕ), ↑v i * h i * e i x
right✝: ∀ (x : ↑Ω), Summable fun i => ↑v i * h i * e i x
x: ↑Ω
comm: ∀ (i : ℕ), ↑v i * a * h i * e i x = a * ↑v i * h i * e i x
h
g x = ∑' (i : ℕ), a * ↑v i * h i * e i x

      
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
x: ↑Ω
right✝¹: Summable fun i => ↑v i * h i ^ 2
f_repr: ↑f = fun x => ∑' (i : ℕ), ↑v i * h i * e i x
right✝: ∀ (x : ↑Ω), Summable fun i => ↑v i * h i * e i x
g = fun x => ∑' (i : ℕ), ↑v i * a * h i * e i xhave summand_comm : ∀ i, a * v.1 i * h i * e i x = a * (v.1 i * h i * e i x) := 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
x✝: ↑Ω
right✝¹: Summable fun i => ↑v i * h i ^ 2
f_repr: ↑f = fun x => ∑' (i : ℕ), ↑v i * h i * e i x
right✝: ∀ (x : ↑Ω), Summable fun i => ↑v i * h i * e i x
x: ↑Ω
comm: ∀ (i : ℕ), ↑v i * a * h i * e i x = a * ↑v i * h i * e i x
h
g x = ∑' (i : ℕ), a * ↑v i * h i * e i xby
Goals accomplished! 🐙 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
x✝: ↑Ω
right✝¹: Summable fun i => ↑v i * h i ^ 2
f_repr: ↑f = fun x => ∑' (i : ℕ), ↑v i * h i * e i x
right✝: ∀ (x : ↑Ω), Summable fun i => ↑v i * h i * e i x
x: ↑Ω
comm: ∀ (i : ℕ), ↑v i * a * h i * e i x = a * ↑v i * h i * e i x
∀ (i : ℕ), a * ↑v i * h i * e i x = a * (↑v i * h i * e i x){
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
x✝: ↑Ω
right✝¹: Summable fun i => ↑v i * h i ^ 2
f_repr: ↑f = fun x => ∑' (i : ℕ), ↑v i * h i * e i x
right✝: ∀ (x : ↑Ω), Summable fun i => ↑v i * h i * e i x
x: ↑Ω
comm: ∀ (i : ℕ), ↑v i * a * h i * e i x = a * ↑v i * h i * e i x
∀ (i : ℕ), a * ↑v i * h i * e i x = a * (↑v i * h i * e i x)
        
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
x✝: ↑Ω
right✝¹: Summable fun i => ↑v i * h i ^ 2
f_repr: ↑f = fun x => ∑' (i : ℕ), ↑v i * h i * e i x
right✝: ∀ (x : ↑Ω), Summable fun i => ↑v i * h i * e i x
x: ↑Ω
comm: ∀ (i : ℕ), ↑v i * a * h i * e i x = a * ↑v i * h i * e i x
∀ (i : ℕ), a * ↑v i * h i * e i x = a * (↑v i * h i * e i x)intro i
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
x✝: ↑Ω
right✝¹: Summable fun i => ↑v i * h i ^ 2
f_repr: ↑f = fun x => ∑' (i : ℕ), ↑v i * h i * e i x
right✝: ∀ (x : ↑Ω), Summable fun i => ↑v i * h i * e i x
x: ↑Ω
comm: ∀ (i : ℕ), ↑v i * a * h i * e i x = a * ↑v i * h i * e i x
i: ℕ
a * ↑v i * h i * e i x = a * (↑v i * h i * e i x)
        
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
x✝: ↑Ω
right✝¹: Summable fun i => ↑v i * h i ^ 2
f_repr: ↑f = fun x => ∑' (i : ℕ), ↑v i * h i * e i x
right✝: ∀ (x : ↑Ω), Summable fun i => ↑v i * h i * e i x
x: ↑Ω
comm: ∀ (i : ℕ), ↑v i * a * h i * e i x = a * ↑v i * h i * e i x
∀ (i : ℕ), a * ↑v i * h i * e i x = a * (↑v i * h i * e i x)ring
Goals accomplished! 🐙
      }
Goals accomplished! 🐙
      
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
x: ↑Ω
right✝¹: Summable fun i => ↑v i * h i ^ 2
f_repr: ↑f = fun x => ∑' (i : ℕ), ↑v i * h i * e i x
right✝: ∀ (x : ↑Ω), Summable fun i => ↑v i * h i * e i x
g = fun x => ∑' (i : ℕ), ↑v i * a * h i * e i xsimp_rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
x✝: ↑Ω
right✝¹: Summable fun i => ↑v i * h i ^ 2
f_repr: ↑f = fun x => ∑' (i : ℕ), ↑v i * h i * e i x
right✝: ∀ (x : ↑Ω), Summable fun i => ↑v i * h i * e i x
x: ↑Ω
comm: ∀ (i : ℕ), ↑v i * a * h i * e i x = a * ↑v i * h i * e i x
summand_comm: ∀ (i : ℕ), a * ↑v i * h i * e i x = a * (↑v i * h i * e i x)
h
g x = ∑' (i : ℕ), a * ↑v i * h i * e i xsummand_comm
[Meta.Tactic.simp.rewrite] summand_comm:1000, a * ↑v i * h i * e i x ==> a * (↑v i * h i * e i x)
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
x✝: ↑Ω
right✝¹: Summable fun i => ↑v i * h i ^ 2
f_repr: ↑f = fun x => ∑' (i : ℕ), ↑v i * h i * e i x
right✝: ∀ (x : ↑Ω), Summable fun i => ↑v i * h i * e i x
x: ↑Ω
comm: ∀ (i : ℕ), ↑v i * a * h i * e i x = a * ↑v i * h i * e i x
summand_comm: ∀ (i : ℕ), a * ↑v i * h i * e i x = a * (↑v i * h i * e i x)
h
g x = ∑' (i : ℕ), a * (↑v i * h i * e i x)]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
x✝: ↑Ω
right✝¹: Summable fun i => ↑v i * h i ^ 2
f_repr: ↑f = fun x => ∑' (i : ℕ), ↑v i * h i * e i x
right✝: ∀ (x : ↑Ω), Summable fun i => ↑v i * h i * e i x
x: ↑Ω
comm: ∀ (i : ℕ), ↑v i * a * h i * e i x = a * ↑v i * h i * e i x
summand_comm: ∀ (i : ℕ), a * ↑v i * h i * e i x = a * (↑v i * h i * e i x)
h
g x = ∑' (i : ℕ), a * (↑v i * h i * e i x)

      
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
x: ↑Ω
right✝¹: Summable fun i => ↑v i * h i ^ 2
f_repr: ↑f = fun x => ∑' (i : ℕ), ↑v i * h i * e i x
right✝: ∀ (x : ↑Ω), Summable fun i => ↑v i * h i * e i x
g = fun x => ∑' (i : ℕ), ↑v i * a * h i * e i xhave const_out : ∑' i, a * (λ i ↦ (v.1 i * h i * e i x)) i = a * ∑' i, (λ i ↦ (v.1 i * h i * e i x)) i := 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
x✝: ↑Ω
right✝¹: Summable fun i => ↑v i * h i ^ 2
f_repr: ↑f = fun x => ∑' (i : ℕ), ↑v i * h i * e i x
right✝: ∀ (x : ↑Ω), Summable fun i => ↑v i * h i * e i x
x: ↑Ω
comm: ∀ (i : ℕ), ↑v i * a * h i * e i x = a * ↑v i * h i * e i x
summand_comm: ∀ (i : ℕ), a * ↑v i * h i * e i x = a * (↑v i * h i * e i x)
h
g x = ∑' (i : ℕ), a * (↑v i * h i * e i x)by
Goals accomplished! 🐙 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
x✝: ↑Ω
right✝¹: Summable fun i => ↑v i * h i ^ 2
f_repr: ↑f = fun x => ∑' (i : ℕ), ↑v i * h i * e i x
right✝: ∀ (x : ↑Ω), Summable fun i => ↑v i * h i * e i x
x: ↑Ω
comm: ∀ (i : ℕ), ↑v i * a * h i * e i x = a * ↑v i * h i * e i x
summand_comm: ∀ (i : ℕ), a * ↑v i * h i * e i x = a * (↑v i * h i * e i x)
∑' (i : ℕ), a * (fun i => ↑v i * h i * e i x) i = a * ∑' (i : ℕ), (fun i => ↑v i * h i * e i x) iexact tsum_mul_left
Goals accomplished! 🐙
      
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
x: ↑Ω
right✝¹: Summable fun i => ↑v i * h i ^ 2
f_repr: ↑f = fun x => ∑' (i : ℕ), ↑v i * h i * e i x
right✝: ∀ (x : ↑Ω), Summable fun i => ↑v i * h i * e i x
g = fun x => ∑' (i : ℕ), ↑v i * a * h i * e i xrw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
x✝: ↑Ω
right✝¹: Summable fun i => ↑v i * h i ^ 2
f_repr: ↑f = fun x => ∑' (i : ℕ), ↑v i * h i * e i x
right✝: ∀ (x : ↑Ω), Summable fun i => ↑v i * h i * e i x
x: ↑Ω
comm: ∀ (i : ℕ), ↑v i * a * h i * e i x = a * ↑v i * h i * e i x
summand_comm: ∀ (i : ℕ), a * ↑v i * h i * e i x = a * (↑v i * h i * e i x)
const_out: ∑' (i : ℕ), a * (fun i => ↑v i * h i * e i x) i = a * ∑' (i : ℕ), (fun i => ↑v i * h i * e i x) i
h
g x = ∑' (i : ℕ), a * (↑v i * h i * e i x)const_out,
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
x✝: ↑Ω
right✝¹: Summable fun i => ↑v i * h i ^ 2
f_repr: ↑f = fun x => ∑' (i : ℕ), ↑v i * h i * e i x
right✝: ∀ (x : ↑Ω), Summable fun i => ↑v i * h i * e i x
x: ↑Ω
comm: ∀ (i : ℕ), ↑v i * a * h i * e i x = a * ↑v i * h i * e i x
summand_comm: ∀ (i : ℕ), a * ↑v i * h i * e i x = a * (↑v i * h i * e i x)
const_out: ∑' (i : ℕ), a * (fun i => ↑v i * h i * e i x) i = a * ∑' (i : ℕ), (fun i => ↑v i * h i * e i x) i
h
g x = a * ∑' (i : ℕ), (fun i => ↑v i * h i * e i x) i 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
x✝: ↑Ω
right✝¹: Summable fun i => ↑v i * h i ^ 2
f_repr: ↑f = fun x => ∑' (i : ℕ), ↑v i * h i * e i x
right✝: ∀ (x : ↑Ω), Summable fun i => ↑v i * h i * e i x
x: ↑Ω
comm: ∀ (i : ℕ), ↑v i * a * h i * e i x = a * ↑v i * h i * e i x
summand_comm: ∀ (i : ℕ), a * ↑v i * h i * e i x = a * (↑v i * h i * e i x)
const_out: ∑' (i : ℕ), a * (fun i => ↑v i * h i * e i x) i = a * ∑' (i : ℕ), (fun i => ↑v i * h i * e i x) i
h
g x = ∑' (i : ℕ), a * (↑v i * h i * e i x)←congrFun f_repr x
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
x✝: ↑Ω
right✝¹: Summable fun i => ↑v i * h i ^ 2
f_repr: ↑f = fun x => ∑' (i : ℕ), ↑v i * h i * e i x
right✝: ∀ (x : ↑Ω), Summable fun i => ↑v i * h i * e i x
x: ↑Ω
comm: ∀ (i : ℕ), ↑v i * a * h i * e i x = a * ↑v i * h i * e i x
summand_comm: ∀ (i : ℕ), a * ↑v i * h i * e i x = a * (↑v i * h i * e i x)
const_out: ∑' (i : ℕ), a * (fun i => ↑v i * h i * e i x) i = a * ∑' (i : ℕ), (fun i => ↑v i * h i * e i x) i
h
g x = a * ↑f x]
Goals accomplished! 🐙
    }
Goals accomplished! 🐙
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
f_repr: h ∈ set_repr f
left
(fun x => a * ↑f x) = fun x => ∑' (i : ℕ), ↑v i * (fun i => a * Set.Nonempty.some ⋯ i) i * e i xhave g_x := congrFun g_eq_tsum x
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
x: ↑Ω
right✝¹: Summable fun i => ↑v i * h i ^ 2
f_repr: ↑f = fun x => ∑' (i : ℕ), ↑v i * h i * e i x
right✝: ∀ (x : ↑Ω), Summable fun i => ↑v i * h i * e i x
g_eq_tsum: g = fun x => ∑' (i : ℕ), ↑v i * a * h i * e i x
g_x: g x = ∑' (i : ℕ), ↑v i * a * h i * e i x
left.h.intro.intro
a * ↑f x = ∑' (i : ℕ), ↑v i * (fun i => a * Set.Nonempty.some ⋯ i) i * e i x
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
f_repr: h ∈ set_repr f
left
(fun x => a * ↑f x) = fun x => ∑' (i : ℕ), ↑v i * (fun i => a * Set.Nonempty.some ⋯ i) i * e i xsimp_rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
x: ↑Ω
right✝¹: Summable fun i => ↑v i * h i ^ 2
f_repr: ↑f = fun x => ∑' (i : ℕ), ↑v i * h i * e i x
right✝: ∀ (x : ↑Ω), Summable fun i => ↑v i * h i * e i x
g_eq_tsum: g = fun x => ∑' (i : ℕ), ↑v i * a * h i * e i x
g_x: g x = ∑' (i : ℕ), ↑v i * a * h i * e i x
left.h.intro.intro
a * ↑f x = ∑' (i : ℕ), ↑v i * (fun i => a * Set.Nonempty.some ⋯ i) i * e i xshow ∀ i, v.1 i * a * h i * e i x = v.1 i * (a * h i) * e i x by 
[Meta.Tactic.simp.rewrite] show ∀ i, v.1 i * a * h i * e i x = v.1 i * (a * h i) * e i x by intro i;
        ring:1000, ↑v i * a * h i * e i x ==> ↑v i * (a * h i) * e i x
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
x: ↑Ω
right✝¹: Summable fun i => ↑v i * h i ^ 2
f_repr: ↑f = fun x => ∑' (i : ℕ), ↑v i * h i * e i x
right✝: ∀ (x : ↑Ω), Summable fun i => ↑v i * h i * e i x
g_eq_tsum: g = fun x => ∑' (i : ℕ), ↑v i * a * h i * e i x
g_x: g x = ∑' (i : ℕ), ↑v i * a * h i * e i x
left.h.intro.intro
a * ↑f x = ∑' (i : ℕ), ↑v i * (fun i => a * Set.Nonempty.some ⋯ i) i * e i xintro i
[Meta.Tactic.simp.rewrite] show ∀ i, v.1 i * a * h i * e i x = v.1 i * (a * h i) * e i x by intro i;
        ring:1000, ↑v i * a * h i * e i x ==> ↑v i * (a * h i) * e i x
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
x: ↑Ω
right✝¹: Summable fun i => ↑v i * h i ^ 2
f_repr: ↑f = fun x => ∑' (i : ℕ), ↑v i * h i * e i x
right✝: ∀ (x : ↑Ω), Summable fun i => ↑v i * h i * e i x
g_eq_tsum: g = fun x => ∑' (i : ℕ), ↑v i * a * h i * e i x
g_x: g x = ∑' (i : ℕ), ↑v i * a * h i * e i x
i: ℕ
↑v i * a * h i * e i x = ↑v i * (a * h i) * e i x;
[Meta.Tactic.simp.rewrite] show ∀ i, v.1 i * a * h i * e i x = v.1 i * (a * h i) * e i x by intro i;
        ring:1000, ↑v i * a * h i * e i x ==> ↑v i * (a * h i) * e i x
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
x: ↑Ω
right✝¹: Summable fun i => ↑v i * h i ^ 2
f_repr: ↑f = fun x => ∑' (i : ℕ), ↑v i * h i * e i x
right✝: ∀ (x : ↑Ω), Summable fun i => ↑v i * h i * e i x
g_eq_tsum: g = fun x => ∑' (i : ℕ), ↑v i * a * h i * e i x
g_x: g x = ∑' (i : ℕ), ↑v i * a * h i * e i x
i: ℕ
↑v i * a * h i * e i x = ↑v i * (a * h i) * e i x 
[Meta.Tactic.simp.rewrite] show ∀ i, v.1 i * a * h i * e i x = v.1 i * (a * h i) * e i x by intro i;
        ring:1000, ↑v i * a * h i * e i x ==> ↑v i * (a * h i) * e i x
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
x: ↑Ω
right✝¹: Summable fun i => ↑v i * h i ^ 2
f_repr: ↑f = fun x => ∑' (i : ℕ), ↑v i * h i * e i x
right✝: ∀ (x : ↑Ω), Summable fun i => ↑v i * h i * e i x
g_eq_tsum: g = fun x => ∑' (i : ℕ), ↑v i * a * h i * e i x
g_x: g x = ∑' (i : ℕ), ↑v i * a * h i * e i x
∀ (i : ℕ), ↑v i * a * h i * e i x = ↑v i * (a * h i) * e i xring
[Meta.Tactic.simp.rewrite] show ∀ i, v.1 i * a * h i * e i x = v.1 i * (a * h i) * e i x by intro i;
        ring:1000, ↑v i * a * h i * e i x ==> ↑v i * (a * h i) * e i x
Goals accomplished! 🐙] at g_x
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
x: ↑Ω
right✝¹: Summable fun i => ↑v i * h i ^ 2
f_repr: ↑f = fun x => ∑' (i : ℕ), ↑v i * h i * e i x
right✝: ∀ (x : ↑Ω), Summable fun i => ↑v i * h i * e i x
g_eq_tsum: g = fun x => ∑' (i : ℕ), ↑v i * a * h i * e i x
g_x: g x = ∑' (i : ℕ), ↑v i * (a * h i) * e i x
left.h.intro.intro
a * ↑f x = ∑' (i : ℕ), ↑v i * (fun i => a * Set.Nonempty.some ⋯ i) i * e i x
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
f_repr: h ∈ set_repr f
left
(fun x => a * ↑f x) = fun x => ∑' (i : ℕ), ↑v i * (fun i => a * Set.Nonempty.some ⋯ i) i * e i xsimp_rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
x: ↑Ω
right✝¹: Summable fun i => ↑v i * h i ^ 2
f_repr: ↑f = fun x => ∑' (i : ℕ), ↑v i * h i * e i x
right✝: ∀ (x : ↑Ω), Summable fun i => ↑v i * h i * e i x
g_eq_tsum: g = fun x => ∑' (i : ℕ), ↑v i * a * h i * e i x
g_x: g x = ∑' (i : ℕ), ↑v i * (a * h i) * e i x
left.h.intro.intro
a * ↑f x = ∑' (i : ℕ), ↑v i * (fun i => a * Set.Nonempty.some ⋯ i) i * e i xshow ∀ i, a * h i = g_h i by 
[Meta.Tactic.simp.rewrite] show ∀ i, a * h i = g_h i by intro i; rfl:1000, a * h i ==> g_h i
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
x: ↑Ω
right✝¹: Summable fun i => ↑v i * h i ^ 2
f_repr: ↑f = fun x => ∑' (i : ℕ), ↑v i * h i * e i x
right✝: ∀ (x : ↑Ω), Summable fun i => ↑v i * h i * e i x
g_eq_tsum: g = fun x => ∑' (i : ℕ), ↑v i * a * h i * e i x
g_x: g x = ∑' (i : ℕ), ↑v i * (a * h i) * e i x
left.h.intro.intro
a * ↑f x = ∑' (i : ℕ), ↑v i * (fun i => a * Set.Nonempty.some ⋯ i) i * e i xintro i
[Meta.Tactic.simp.rewrite] show ∀ i, a * h i = g_h i by intro i; rfl:1000, a * h i ==> g_h i
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
x: ↑Ω
right✝¹: Summable fun i => ↑v i * h i ^ 2
f_repr: ↑f = fun x => ∑' (i : ℕ), ↑v i * h i * e i x
right✝: ∀ (x : ↑Ω), Summable fun i => ↑v i * h i * e i x
g_eq_tsum: g = fun x => ∑' (i : ℕ), ↑v i * a * h i * e i x
g_x: g x = ∑' (i : ℕ), ↑v i * (a * h i) * e i x
i: ℕ
a * h i = g_h i;
[Meta.Tactic.simp.rewrite] show ∀ i, a * h i = g_h i by intro i; rfl:1000, a * h i ==> g_h i
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
x: ↑Ω
right✝¹: Summable fun i => ↑v i * h i ^ 2
f_repr: ↑f = fun x => ∑' (i : ℕ), ↑v i * h i * e i x
right✝: ∀ (x : ↑Ω), Summable fun i => ↑v i * h i * e i x
g_eq_tsum: g = fun x => ∑' (i : ℕ), ↑v i * a * h i * e i x
g_x: g x = ∑' (i : ℕ), ↑v i * (a * h i) * e i x
i: ℕ
a * h i = g_h i 
[Meta.Tactic.simp.rewrite] show ∀ i, a * h i = g_h i by intro i; rfl:1000, a * h i ==> g_h i
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
x: ↑Ω
right✝¹: Summable fun i => ↑v i * h i ^ 2
f_repr: ↑f = fun x => ∑' (i : ℕ), ↑v i * h i * e i x
right✝: ∀ (x : ↑Ω), Summable fun i => ↑v i * h i * e i x
g_eq_tsum: g = fun x => ∑' (i : ℕ), ↑v i * a * h i * e i x
g_x: g x = ∑' (i : ℕ), ↑v i * (a * h i) * e i x
∀ (i : ℕ), a * h i = g_h irfl
[Meta.Tactic.simp.rewrite] show ∀ i, a * h i = g_h i by intro i; rfl:1000, a * h i ==> g_h i
Goals accomplished! 🐙] at g_x
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
x: ↑Ω
right✝¹: Summable fun i => ↑v i * h i ^ 2
f_repr: ↑f = fun x => ∑' (i : ℕ), ↑v i * h i * e i x
right✝: ∀ (x : ↑Ω), Summable fun i => ↑v i * h i * e i x
g_eq_tsum: g = fun x => ∑' (i : ℕ), ↑v i * a * h i * e i x
g_x: g x = ∑' (i : ℕ), ↑v i * g_h i * e i x
left.h.intro.intro
a * ↑f x = ∑' (i : ℕ), ↑v i * (fun i => a * Set.Nonempty.some ⋯ i) i * e i x
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
f_repr: h ∈ set_repr f
left
(fun x => a * ↑f x) = fun x => ∑' (i : ℕ), ↑v i * (fun i => a * Set.Nonempty.some ⋯ i) i * e i xexact g_x
Goals accomplished! 🐙
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
f_repr v e (fun x => a * ↑f x) fun i => a * Set.Nonempty.some ⋯ iintro x
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
f_repr: h ∈ set_repr f
x: ↑Ω
right
Summable fun i => ↑v i * (fun i => a * Set.Nonempty.some ⋯ i) i * e i x
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
f_repr v e (fun x => a * ↑f x) fun i => a * Set.Nonempty.some ⋯ ihave remove_function : (λ i ↦ v.1 i * (λ i ↦ a * h i) i * e i x) =(λ i ↦ a * (v.1 i * h i * e i x)) := 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
f_repr: h ∈ set_repr f
x: ↑Ω
right
Summable fun i => ↑v i * (fun i => a * Set.Nonempty.some ⋯ i) i * e i xby
Goals accomplished! 🐙 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
f_repr: h ∈ set_repr f
x: ↑Ω
(fun i => ↑v i * (fun i => a * h i) i * e i x) = fun i => a * (↑v i * h i * e i x){
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
f_repr: h ∈ set_repr f
x: ↑Ω
(fun i => ↑v i * (fun i => a * h i) i * e i x) = fun i => a * (↑v i * h i * e i x)
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
f_repr: h ∈ set_repr f
x: ↑Ω
(fun i => ↑v i * (fun i => a * h i) i * e i x) = fun i => a * (↑v i * h i * e i x)ext i
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
f_repr: h ∈ set_repr f
x: ↑Ω
i: ℕ
h
↑v i * (fun i => a * h i) i * e i x = a * (↑v i * h i * e i x)
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
f_repr: h ∈ set_repr f
x: ↑Ω
(fun i => ↑v i * (fun i => a * h i) i * e i x) = fun i => a * (↑v i * h i * e i x)ring
Goals accomplished! 🐙
  }
Goals accomplished! 🐙
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
f_repr v e (fun x => a * ↑f x) fun i => a * Set.Nonempty.some ⋯ irw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
f_repr: h ∈ set_repr f
x: ↑Ω
remove_function: (fun i => ↑v i * (fun i => a * h i) i * e i x) = fun i => a * (↑v i * h i * e i x)
right
Summable fun i => ↑v i * (fun i => a * Set.Nonempty.some ⋯ i) i * e i xremove_function
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
f_repr: h ∈ set_repr f
x: ↑Ω
remove_function: (fun i => ↑v i * (fun i => a * h i) i * e i x) = fun i => a * (↑v i * h i * e i x)
right
Summable fun i => a * (↑v i * h i * e i x)]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
f_repr: h ∈ set_repr f
x: ↑Ω
remove_function: (fun i => ↑v i * (fun i => a * h i) i * e i x) = fun i => a * (↑v i * h i * e i x)
right
Summable fun i => a * (↑v i * h i * e i x)
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
f_repr v e (fun x => a * ↑f x) fun i => a * Set.Nonempty.some ⋯ iexact (f_repr.1.2 x).mul_left a
Goals accomplished! 🐙

lemma mul_summable (a : ℝ) : Summable (λ i ↦ v.1 i * (λ i ↦ a * (set_repr_ne f).some i) i ^ 2) := by
Goals accomplished! 🐙
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
Summable fun i => ↑v i * (fun i => a * Set.Nonempty.some ⋯ i) i ^ 2let h := (set_repr_ne f).some
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
Summable fun i => ↑v i * (fun i => a * Set.Nonempty.some ⋯ i) i ^ 2
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
Summable fun i => ↑v i * (fun i => a * Set.Nonempty.some ⋯ i) i ^ 2let g_h := λ i ↦ a * h i
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
Summable fun i => ↑v i * (fun i => a * Set.Nonempty.some ⋯ i) i ^ 2
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
Summable fun i => ↑v i * (fun i => a * Set.Nonempty.some ⋯ i) i ^ 2have set_repr : h ∈ set_repr f := (set_repr_ne f).some_mem
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
set_repr: h ∈ _root_.set_repr f
Summable fun i => ↑v i * (fun i => a * Set.Nonempty.some ⋯ i) i ^ 2

  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
Summable fun i => ↑v i * (fun i => a * Set.Nonempty.some ⋯ i) i ^ 2rcases set_repr with ⟨_, h_summable⟩
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
left✝: f_repr v e (↑f) h
h_summable: Summable fun i => ↑v i * h i ^ 2
intro
Summable fun i => ↑v i * (fun i => a * Set.Nonempty.some ⋯ i) i ^ 2
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
Summable fun i => ↑v i * (fun i => a * Set.Nonempty.some ⋯ i) i ^ 2have comm_fun : (λ i ↦ v.1 i * (g_h i)^2) = (λ i ↦ a^2 * (v.1 i * (h i)^2)) := 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
left✝: f_repr v e (↑f) h
h_summable: Summable fun i => ↑v i * h i ^ 2
intro
Summable fun i => ↑v i * (fun i => a * Set.Nonempty.some ⋯ i) i ^ 2by
Goals accomplished! 🐙 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
left✝: f_repr v e (↑f) h
h_summable: Summable fun i => ↑v i * h i ^ 2
(fun i => ↑v i * g_h i ^ 2) = fun i => a ^ 2 * (↑v i * h i ^ 2){
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
left✝: f_repr v e (↑f) h
h_summable: Summable fun i => ↑v i * h i ^ 2
(fun i => ↑v i * g_h i ^ 2) = fun i => a ^ 2 * (↑v i * h i ^ 2)
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
left✝: f_repr v e (↑f) h
h_summable: Summable fun i => ↑v i * h i ^ 2
(fun i => ↑v i * g_h i ^ 2) = fun i => a ^ 2 * (↑v i * h i ^ 2)ext i
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
left✝: f_repr v e (↑f) h
h_summable: Summable fun i => ↑v i * h i ^ 2
i: ℕ
h
↑v i * g_h i ^ 2 = a ^ 2 * (↑v i * h i ^ 2)
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
left✝: f_repr v e (↑f) h
h_summable: Summable fun i => ↑v i * h i ^ 2
(fun i => ↑v i * g_h i ^ 2) = fun i => a ^ 2 * (↑v i * h i ^ 2)rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
left✝: f_repr v e (↑f) h
h_summable: Summable fun i => ↑v i * h i ^ 2
i: ℕ
h
↑v i * g_h i ^ 2 = a ^ 2 * (↑v i * h i ^ 2)show g_h i = a * h i by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
left✝: f_repr v e (↑f) h
h_summable: Summable fun i => ↑v i * h i ^ 2
i: ℕ
h
↑v i * g_h i ^ 2 = a ^ 2 * (↑v i * h i ^ 2)rfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
left✝: f_repr v e (↑f) h
h_summable: Summable fun i => ↑v i * h i ^ 2
i: ℕ
h
↑v i * (a * h i) ^ 2 = a ^ 2 * (↑v i * h i ^ 2)
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
left✝: f_repr v e (↑f) h
h_summable: Summable fun i => ↑v i * h i ^ 2
(fun i => ↑v i * g_h i ^ 2) = fun i => a ^ 2 * (↑v i * h i ^ 2)rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
left✝: f_repr v e (↑f) h
h_summable: Summable fun i => ↑v i * h i ^ 2
i: ℕ
h
↑v i * (a * h i) ^ 2 = a ^ 2 * (↑v i * h i ^ 2)mul_pow a (h i) 2
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
left✝: f_repr v e (↑f) h
h_summable: Summable fun i => ↑v i * h i ^ 2
i: ℕ
h
↑v i * (a ^ 2 * h i ^ 2) = a ^ 2 * (↑v i * h i ^ 2)]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
left✝: f_repr v e (↑f) h
h_summable: Summable fun i => ↑v i * h i ^ 2
i: ℕ
h
↑v i * (a ^ 2 * h i ^ 2) = a ^ 2 * (↑v i * h i ^ 2)
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
left✝: f_repr v e (↑f) h
h_summable: Summable fun i => ↑v i * h i ^ 2
(fun i => ↑v i * g_h i ^ 2) = fun i => a ^ 2 * (↑v i * h i ^ 2)rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
left✝: f_repr v e (↑f) h
h_summable: Summable fun i => ↑v i * h i ^ 2
i: ℕ
h
↑v i * (a ^ 2 * h i ^ 2) = a ^ 2 * (↑v i * h i ^ 2)show (v.1 i) * (a^2 * (h i)^2) = (a^2) * (v.1 i * (h i)^2) by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
left✝: f_repr v e (↑f) h
h_summable: Summable fun i => ↑v i * h i ^ 2
i: ℕ
h
↑v i * (a ^ 2 * h i ^ 2) = a ^ 2 * (↑v i * h i ^ 2)ring
Goals accomplished! 🐙]
Goals accomplished! 🐙

  }
Goals accomplished! 🐙
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
Summable fun i => ↑v i * (fun i => a * Set.Nonempty.some ⋯ i) i ^ 2rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
left✝: f_repr v e (↑f) h
h_summable: Summable fun i => ↑v i * h i ^ 2
comm_fun: (fun i => ↑v i * g_h i ^ 2) = fun i => a ^ 2 * (↑v i * h i ^ 2)
intro
Summable fun i => ↑v i * (fun i => a * Set.Nonempty.some ⋯ i) i ^ 2comm_fun
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
left✝: f_repr v e (↑f) h
h_summable: Summable fun i => ↑v i * h i ^ 2
comm_fun: (fun i => ↑v i * g_h i ^ 2) = fun i => a ^ 2 * (↑v i * h i ^ 2)
intro
Summable fun i => a ^ 2 * (↑v i * h i ^ 2)]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
left✝: f_repr v e (↑f) h
h_summable: Summable fun i => ↑v i * h i ^ 2
comm_fun: (fun i => ↑v i * g_h i ^ 2) = fun i => a ^ 2 * (↑v i * h i ^ 2)
intro
Summable fun i => a ^ 2 * (↑v i * h i ^ 2)
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
Summable fun i => ↑v i * (fun i => a * Set.Nonempty.some ⋯ i) i ^ 2exact (h_summable.mul_left (a^2))
Goals accomplished! 🐙

lemma mul_in_H (a : ℝ) : (λ x ↦ a * f.1 x) ∈ (H v e μ) := by
Goals accomplished! 🐙
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
(fun x => a * ↑f x) ∈ H v e μlet g := λ x ↦ a * f.1 x
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
(fun x => a * ↑f x) ∈ H v e μ
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
(fun x => a * ↑f x) ∈ H v e μhave g_in_L2 : g ∈ L2 μ := Memℒp.const_mul (f.2).1 a
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
g_in_L2: g ∈ L2 μ
(fun x => a * ↑f x) ∈ H v e μ
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
(fun x => a * ↑f x) ∈ H v e μlet h := (set_repr_ne f).some
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
g_in_L2: g ∈ L2 μ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
(fun x => a * ↑f x) ∈ H v e μ
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
(fun x => a * ↑f x) ∈ H v e μlet g_h := λ i ↦ a * h i
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g✝: ↑(H v e μ)
a: ℝ
g:= fun x => a * ↑f x: ↑Ω → ℝ
g_in_L2: g ∈ L2 μ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
g_h:= fun i => a * h i: ℕ → ℝ
(fun x => a * ↑f x) ∈ H v e μ
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
(fun x => a * ↑f x) ∈ H v e μrefine ⟨g_in_L2, g_h, mul_repr f a, mul_summable f a⟩
Goals accomplished! 🐙

instance : HMul ℝ (H v e μ) (H v e μ) where
  hMul := λ a f ↦ ⟨λ x ↦ a * f.1 x, mul_in_H f a⟩

instance : HSMul ℝ (H v e μ) (H v e μ) where
  hSMul := λ r f ↦ r * f

end Ring

/-
  We define the sum between two functions in H as the pointwise sum. We show that the result lies in H. We also define the 0 function of H. We show several properties on the addition.
-/

namespace Group

variable {v : eigen} {e : ℕ → Ω → ℝ} {μ : Measure Ω} [IsFiniteMeasure μ] (f g : H v e μ)

lemma add_in_L2 : (λ x ↦ f.1 x + g.1 x) ∈ L2 μ := Memℒp.add (f.2.1) (g.2.1)

lemma add_summable : Summable (λ i ↦ v.1 i * ((set_repr_ne f).some i + (set_repr_ne g).some i)^2) := by
Goals accomplished! 🐙
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
Summable fun i => ↑v i * (Set.Nonempty.some ⋯ i + Set.Nonempty.some ⋯ i) ^ 2let a_f := (set_repr_ne f).some
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
Summable fun i => ↑v i * (Set.Nonempty.some ⋯ i + Set.Nonempty.some ⋯ i) ^ 2
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
Summable fun i => ↑v i * (Set.Nonempty.some ⋯ i + Set.Nonempty.some ⋯ i) ^ 2let a_g := (set_repr_ne g).some
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
Summable fun i => ↑v i * (Set.Nonempty.some ⋯ i + Set.Nonempty.some ⋯ i) ^ 2
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
Summable fun i => ↑v i * (Set.Nonempty.some ⋯ i + Set.Nonempty.some ⋯ i) ^ 2have f_eq : (λ i ↦ v.1 i * (a_f i + a_g i)^2) = (λ i ↦ v.1 i * a_f i * a_f i + 2 * (v.1 i * a_f i * a_g i) + v.1 i * a_g i * a_g i) := 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
Summable fun i => ↑v i * (Set.Nonempty.some ⋯ i + Set.Nonempty.some ⋯ i) ^ 2by
Goals accomplished! 🐙 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
(fun i => ↑v i * (a_f i + a_g i) ^ 2) = fun i =>
  ↑v i * a_f i * a_f i + 2 * (↑v i * a_f i * a_g i) + ↑v i * a_g i * a_g i{
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
(fun i => ↑v i * (a_f i + a_g i) ^ 2) = fun i =>
  ↑v i * a_f i * a_f i + 2 * (↑v i * a_f i * a_g i) + ↑v i * a_g i * a_g i
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
(fun i => ↑v i * (a_f i + a_g i) ^ 2) = fun i =>
  ↑v i * a_f i * a_f i + 2 * (↑v i * a_f i * a_g i) + ↑v i * a_g i * a_g iext i
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
i: ℕ
h
↑v i * (a_f i + a_g i) ^ 2 = ↑v i * a_f i * a_f i + 2 * (↑v i * a_f i * a_g i) + ↑v i * a_g i * a_g i
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
(fun i => ↑v i * (a_f i + a_g i) ^ 2) = fun i =>
  ↑v i * a_f i * a_f i + 2 * (↑v i * a_f i * a_g i) + ↑v i * a_g i * a_g iring
Goals accomplished! 🐙
  }
Goals accomplished! 🐙
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
Summable fun i => ↑v i * (Set.Nonempty.some ⋯ i + Set.Nonempty.some ⋯ i) ^ 2rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
f_eq: (fun i => ↑v i * (a_f i + a_g i) ^ 2) = fun i =>
  ↑v i * a_f i * a_f i + 2 * (↑v i * a_f i * a_g i) + ↑v i * a_g i * a_g i
Summable fun i => ↑v i * (Set.Nonempty.some ⋯ i + Set.Nonempty.some ⋯ i) ^ 2f_eq
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
f_eq: (fun i => ↑v i * (a_f i + a_g i) ^ 2) = fun i =>
  ↑v i * a_f i * a_f i + 2 * (↑v i * a_f i * a_g i) + ↑v i * a_g i * a_g i
Summable fun i => ↑v i * a_f i * a_f i + 2 * (↑v i * a_f i * a_g i) + ↑v i * a_g i * a_g i]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
f_eq: (fun i => ↑v i * (a_f i + a_g i) ^ 2) = fun i =>
  ↑v i * a_f i * a_f i + 2 * (↑v i * a_f i * a_g i) + ↑v i * a_g i * a_g i
Summable fun i => ↑v i * a_f i * a_f i + 2 * (↑v i * a_f i * a_g i) + ↑v i * a_g i * a_g i

  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
Summable fun i => ↑v i * (Set.Nonempty.some ⋯ i + Set.Nonempty.some ⋯ i) ^ 2exact ((product_summable f f).add ((product_summable f g).mul_left 2)).add (product_summable g g)
Goals accomplished! 🐙

lemma add_repr : f_repr v e (λ x ↦ f.1 x + g.1 x) (λ i ↦ (set_repr_ne f).some i + (set_repr_ne g).some i) := by
Goals accomplished! 🐙
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
f_repr v e (fun x => ↑f x + ↑g x) fun i => Set.Nonempty.some ⋯ i + Set.Nonempty.some ⋯ ilet a_f := (set_repr_ne f).some
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
f_repr v e (fun x => ↑f x + ↑g x) fun i => Set.Nonempty.some ⋯ i + Set.Nonempty.some ⋯ i
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
f_repr v e (fun x => ↑f x + ↑g x) fun i => Set.Nonempty.some ⋯ i + Set.Nonempty.some ⋯ ilet a_g := (set_repr_ne g).some
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
f_repr v e (fun x => ↑f x + ↑g x) fun i => Set.Nonempty.some ⋯ i + Set.Nonempty.some ⋯ i

  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
f_repr v e (fun x => ↑f x + ↑g x) fun i => Set.Nonempty.some ⋯ i + Set.Nonempty.some ⋯ iobtain ⟨af_repr, _⟩ := (set_repr_ne f).some_mem
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
af_repr: f_repr v e (↑f) (Set.Nonempty.some ⋯)
right✝: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
intro
f_repr v e (fun x => ↑f x + ↑g x) fun i => Set.Nonempty.some ⋯ i + Set.Nonempty.some ⋯ i
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
f_repr v e (fun x => ↑f x + ↑g x) fun i => Set.Nonempty.some ⋯ i + Set.Nonempty.some ⋯ iobtain ⟨ag_repr, _⟩ := (set_repr_ne g).some_mem
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
af_repr: f_repr v e (↑f) (Set.Nonempty.some ⋯)
right✝¹: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
ag_repr: f_repr v e (↑g) (Set.Nonempty.some ⋯)
right✝: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
intro.intro
f_repr v e (fun x => ↑f x + ↑g x) fun i => Set.Nonempty.some ⋯ i + Set.Nonempty.some ⋯ i

  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
f_repr v e (fun x => ↑f x + ↑g x) fun i => Set.Nonempty.some ⋯ i + Set.Nonempty.some ⋯ ihave summand_distrib : ∀ x, ∀ i, v.1 i * (a_f i + a_g i) * e i x = v.1 i * a_f i * e i x + v.1 i * a_g i * e i x := 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
af_repr: f_repr v e (↑f) (Set.Nonempty.some ⋯)
right✝¹: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
ag_repr: f_repr v e (↑g) (Set.Nonempty.some ⋯)
right✝: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
intro.intro
f_repr v e (fun x => ↑f x + ↑g x) fun i => Set.Nonempty.some ⋯ i + Set.Nonempty.some ⋯ iby
Goals accomplished! 🐙 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
af_repr: f_repr v e (↑f) (Set.Nonempty.some ⋯)
right✝¹: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
ag_repr: f_repr v e (↑g) (Set.Nonempty.some ⋯)
right✝: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
∀ (x : ↑Ω) (i : ℕ), ↑v i * (a_f i + a_g i) * e i x = ↑v i * a_f i * e i x + ↑v i * a_g i * e i xintro x i
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
af_repr: f_repr v e (↑f) (Set.Nonempty.some ⋯)
right✝¹: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
ag_repr: f_repr v e (↑g) (Set.Nonempty.some ⋯)
right✝: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
x: ↑Ω
i: ℕ
↑v i * (a_f i + a_g i) * e i x = ↑v i * a_f i * e i x + ↑v i * a_g i * e i x;
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
af_repr: f_repr v e (↑f) (Set.Nonempty.some ⋯)
right✝¹: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
ag_repr: f_repr v e (↑g) (Set.Nonempty.some ⋯)
right✝: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
x: ↑Ω
i: ℕ
↑v i * (a_f i + a_g i) * e i x = ↑v i * a_f i * e i x + ↑v i * a_g i * e i x 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
af_repr: f_repr v e (↑f) (Set.Nonempty.some ⋯)
right✝¹: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
ag_repr: f_repr v e (↑g) (Set.Nonempty.some ⋯)
right✝: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
∀ (x : ↑Ω) (i : ℕ), ↑v i * (a_f i + a_g i) * e i x = ↑v i * a_f i * e i x + ↑v i * a_g i * e i xring
Goals accomplished! 🐙

  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
f_repr v e (fun x => ↑f x + ↑g x) fun i => Set.Nonempty.some ⋯ i + Set.Nonempty.some ⋯ iconstructor
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
af_repr: f_repr v e (↑f) (Set.Nonempty.some ⋯)
right✝¹: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
ag_repr: f_repr v e (↑g) (Set.Nonempty.some ⋯)
right✝: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
summand_distrib: ∀ (x : ↑Ω) (i : ℕ), ↑v i * (a_f i + a_g i) * e i x = ↑v i * a_f i * e i x + ↑v i * a_g i * e i x
intro.intro.left
(fun x => ↑f x + ↑g x) = fun x => ∑' (i : ℕ), ↑v i * (fun i => Set.Nonempty.some ⋯ i + Set.Nonempty.some ⋯ i) i * e i x
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
af_repr: f_repr v e (↑f) (Set.Nonempty.some ⋯)
right✝¹: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
ag_repr: f_repr v e (↑g) (Set.Nonempty.some ⋯)
right✝: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
summand_distrib: ∀ (x : ↑Ω) (i : ℕ), ↑v i * (a_f i + a_g i) * e i x = ↑v i * a_f i * e i x + ↑v i * a_g i * e i x
intro.intro.right
∀ (x : ↑Ω), Summable fun i => ↑v i * (fun i => Set.Nonempty.some ⋯ i + Set.Nonempty.some ⋯ i) i * e i x
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
f_repr v e (fun x => ↑f x + ↑g x) fun i => Set.Nonempty.some ⋯ i + Set.Nonempty.some ⋯ i·
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
af_repr: f_repr v e (↑f) (Set.Nonempty.some ⋯)
right✝¹: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
ag_repr: f_repr v e (↑g) (Set.Nonempty.some ⋯)
right✝: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
summand_distrib: ∀ (x : ↑Ω) (i : ℕ), ↑v i * (a_f i + a_g i) * e i x = ↑v i * a_f i * e i x + ↑v i * a_g i * e i x
intro.intro.left
(fun x => ↑f x + ↑g x) = fun x => ∑' (i : ℕ), ↑v i * (fun i => Set.Nonempty.some ⋯ i + Set.Nonempty.some ⋯ i) i * e i x 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
af_repr: f_repr v e (↑f) (Set.Nonempty.some ⋯)
right✝¹: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
ag_repr: f_repr v e (↑g) (Set.Nonempty.some ⋯)
right✝: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
summand_distrib: ∀ (x : ↑Ω) (i : ℕ), ↑v i * (a_f i + a_g i) * e i x = ↑v i * a_f i * e i x + ↑v i * a_g i * e i x
intro.intro.left
(fun x => ↑f x + ↑g x) = fun x => ∑' (i : ℕ), ↑v i * (fun i => Set.Nonempty.some ⋯ i + Set.Nonempty.some ⋯ i) i * e i x
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
af_repr: f_repr v e (↑f) (Set.Nonempty.some ⋯)
right✝¹: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
ag_repr: f_repr v e (↑g) (Set.Nonempty.some ⋯)
right✝: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
summand_distrib: ∀ (x : ↑Ω) (i : ℕ), ↑v i * (a_f i + a_g i) * e i x = ↑v i * a_f i * e i x + ↑v i * a_g i * e i x
intro.intro.right
∀ (x : ↑Ω), Summable fun i => ↑v i * (fun i => Set.Nonempty.some ⋯ i + Set.Nonempty.some ⋯ i) i * e i xext x
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
af_repr: f_repr v e (↑f) (Set.Nonempty.some ⋯)
right✝¹: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
ag_repr: f_repr v e (↑g) (Set.Nonempty.some ⋯)
right✝: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
summand_distrib: ∀ (x : ↑Ω) (i : ℕ), ↑v i * (a_f i + a_g i) * e i x = ↑v i * a_f i * e i x + ↑v i * a_g i * e i x
x: ↑Ω
intro.intro.left.h
↑f x + ↑g x = ∑' (i : ℕ), ↑v i * (fun i => Set.Nonempty.some ⋯ i + Set.Nonempty.some ⋯ i) i * e i x
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
af_repr: f_repr v e (↑f) (Set.Nonempty.some ⋯)
right✝¹: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
ag_repr: f_repr v e (↑g) (Set.Nonempty.some ⋯)
right✝: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
summand_distrib: ∀ (x : ↑Ω) (i : ℕ), ↑v i * (a_f i + a_g i) * e i x = ↑v i * a_f i * e i x + ↑v i * a_g i * e i x
intro.intro.left
(fun x => ↑f x + ↑g x) = fun x => ∑' (i : ℕ), ↑v i * (fun i => Set.Nonempty.some ⋯ i + Set.Nonempty.some ⋯ i) i * e i xsimp_rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
af_repr: f_repr v e (↑f) (Set.Nonempty.some ⋯)
right✝¹: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
ag_repr: f_repr v e (↑g) (Set.Nonempty.some ⋯)
right✝: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
summand_distrib: ∀ (x : ↑Ω) (i : ℕ), ↑v i * (a_f i + a_g i) * e i x = ↑v i * a_f i * e i x + ↑v i * a_g i * e i x
x: ↑Ω
intro.intro.left.h
↑f x + ↑g x = ∑' (i : ℕ), ↑v i * (Set.Nonempty.some ⋯ i + Set.Nonempty.some ⋯ i) * e i xsummand_distrib x
[Meta.Tactic.simp.rewrite] summand_distrib
        x:1000, ↑v i * (Set.Nonempty.some ⋯ i + Set.Nonempty.some ⋯ i) *
      e i x ==> ↑v i * a_f i * e i x + ↑v i * a_g i * e i x
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
af_repr: f_repr v e (↑f) (Set.Nonempty.some ⋯)
right✝¹: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
ag_repr: f_repr v e (↑g) (Set.Nonempty.some ⋯)
right✝: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
summand_distrib: ∀ (x : ↑Ω) (i : ℕ), ↑v i * (a_f i + a_g i) * e i x = ↑v i * a_f i * e i x + ↑v i * a_g i * e i x
x: ↑Ω
intro.intro.left.h
↑f x + ↑g x = ∑' (i : ℕ), (↑v i * a_f i * e i x + ↑v i * a_g i * e i x)]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
af_repr: f_repr v e (↑f) (Set.Nonempty.some ⋯)
right✝¹: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
ag_repr: f_repr v e (↑g) (Set.Nonempty.some ⋯)
right✝: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
summand_distrib: ∀ (x : ↑Ω) (i : ℕ), ↑v i * (a_f i + a_g i) * e i x = ↑v i * a_f i * e i x + ↑v i * a_g i * e i x
x: ↑Ω
intro.intro.left.h
↑f x + ↑g x = ∑' (i : ℕ), (↑v i * a_f i * e i x + ↑v i * a_g i * e i x)

    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
af_repr: f_repr v e (↑f) (Set.Nonempty.some ⋯)
right✝¹: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
ag_repr: f_repr v e (↑g) (Set.Nonempty.some ⋯)
right✝: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
summand_distrib: ∀ (x : ↑Ω) (i : ℕ), ↑v i * (a_f i + a_g i) * e i x = ↑v i * a_f i * e i x + ↑v i * a_g i * e i x
intro.intro.left
(fun x => ↑f x + ↑g x) = fun x => ∑' (i : ℕ), ↑v i * (fun i => Set.Nonempty.some ⋯ i + Set.Nonempty.some ⋯ i) i * e i xrw [
      
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
af_repr: f_repr v e (↑f) (Set.Nonempty.some ⋯)
right✝¹: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
ag_repr: f_repr v e (↑g) (Set.Nonempty.some ⋯)
right✝: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
summand_distrib: ∀ (x : ↑Ω) (i : ℕ), ↑v i * (a_f i + a_g i) * e i x = ↑v i * a_f i * e i x + ↑v i * a_g i * e i x
x: ↑Ω
intro.intro.left.h
↑f x + ↑g x = ∑' (i : ℕ), (↑v i * a_f i * e i x + ↑v i * a_g i * e i x)tsum_add (af_repr.2 x) (ag_repr.2 x),
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
af_repr: f_repr v e (↑f) (Set.Nonempty.some ⋯)
right✝¹: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
ag_repr: f_repr v e (↑g) (Set.Nonempty.some ⋯)
right✝: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
summand_distrib: ∀ (x : ↑Ω) (i : ℕ), ↑v i * (a_f i + a_g i) * e i x = ↑v i * a_f i * e i x + ↑v i * a_g i * e i x
x: ↑Ω
intro.intro.left.h
↑f x + ↑g x = ∑' (b : ℕ), ↑v b * Set.Nonempty.some ⋯ b * e b x + ∑' (b : ℕ), ↑v b * Set.Nonempty.some ⋯ b * e b x
      
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
af_repr: f_repr v e (↑f) (Set.Nonempty.some ⋯)
right✝¹: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
ag_repr: f_repr v e (↑g) (Set.Nonempty.some ⋯)
right✝: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
summand_distrib: ∀ (x : ↑Ω) (i : ℕ), ↑v i * (a_f i + a_g i) * e i x = ↑v i * a_f i * e i x + ↑v i * a_g i * e i x
x: ↑Ω
intro.intro.left.h
↑f x + ↑g x = ∑' (i : ℕ), (↑v i * a_f i * e i x + ↑v i * a_g i * e i x)←congrFun af_repr.1 x,
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
af_repr: f_repr v e (↑f) (Set.Nonempty.some ⋯)
right✝¹: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
ag_repr: f_repr v e (↑g) (Set.Nonempty.some ⋯)
right✝: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
summand_distrib: ∀ (x : ↑Ω) (i : ℕ), ↑v i * (a_f i + a_g i) * e i x = ↑v i * a_f i * e i x + ↑v i * a_g i * e i x
x: ↑Ω
intro.intro.left.h
↑f x + ↑g x = ↑f x + ∑' (b : ℕ), ↑v b * Set.Nonempty.some ⋯ b * e b x
      
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
af_repr: f_repr v e (↑f) (Set.Nonempty.some ⋯)
right✝¹: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
ag_repr: f_repr v e (↑g) (Set.Nonempty.some ⋯)
right✝: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
summand_distrib: ∀ (x : ↑Ω) (i : ℕ), ↑v i * (a_f i + a_g i) * e i x = ↑v i * a_f i * e i x + ↑v i * a_g i * e i x
x: ↑Ω
intro.intro.left.h
↑f x + ↑g x = ∑' (i : ℕ), (↑v i * a_f i * e i x + ↑v i * a_g i * e i x)←congrFun ag_repr.1 x
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
af_repr: f_repr v e (↑f) (Set.Nonempty.some ⋯)
right✝¹: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
ag_repr: f_repr v e (↑g) (Set.Nonempty.some ⋯)
right✝: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
summand_distrib: ∀ (x : ↑Ω) (i : ℕ), ↑v i * (a_f i + a_g i) * e i x = ↑v i * a_f i * e i x + ↑v i * a_g i * e i x
x: ↑Ω
intro.intro.left.h
↑f x + ↑g x = ↑f x + ↑g x
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
af_repr: f_repr v e (↑f) (Set.Nonempty.some ⋯)
right✝¹: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
ag_repr: f_repr v e (↑g) (Set.Nonempty.some ⋯)
right✝: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
summand_distrib: ∀ (x : ↑Ω) (i : ℕ), ↑v i * (a_f i + a_g i) * e i x = ↑v i * a_f i * e i x + ↑v i * a_g i * e i x
x: ↑Ω
intro.intro.left.h
↑f x + ↑g x = ∑' (i : ℕ), (↑v i * a_f i * e i x + ↑v i * a_g i * e i x)]
Goals accomplished! 🐙
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
f_repr v e (fun x => ↑f x + ↑g x) fun i => Set.Nonempty.some ⋯ i + Set.Nonempty.some ⋯ iintro x
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
af_repr: f_repr v e (↑f) (Set.Nonempty.some ⋯)
right✝¹: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
ag_repr: f_repr v e (↑g) (Set.Nonempty.some ⋯)
right✝: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
summand_distrib: ∀ (x : ↑Ω) (i : ℕ), ↑v i * (a_f i + a_g i) * e i x = ↑v i * a_f i * e i x + ↑v i * a_g i * e i x
x: ↑Ω
intro.intro.right
Summable fun i => ↑v i * (fun i => Set.Nonempty.some ⋯ i + Set.Nonempty.some ⋯ i) i * e i x
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
f_repr v e (fun x => ↑f x + ↑g x) fun i => Set.Nonempty.some ⋯ i + Set.Nonempty.some ⋯ isimp_rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
af_repr: f_repr v e (↑f) (Set.Nonempty.some ⋯)
right✝¹: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
ag_repr: f_repr v e (↑g) (Set.Nonempty.some ⋯)
right✝: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
summand_distrib: ∀ (x : ↑Ω) (i : ℕ), ↑v i * (a_f i + a_g i) * e i x = ↑v i * a_f i * e i x + ↑v i * a_g i * e i x
x: ↑Ω
intro.intro.right
Summable fun i => ↑v i * (Set.Nonempty.some ⋯ i + Set.Nonempty.some ⋯ i) * e i xsummand_distrib x
[Meta.Tactic.simp.rewrite] summand_distrib
        x:1000, ↑v i * (Set.Nonempty.some ⋯ i + Set.Nonempty.some ⋯ i) *
      e i x ==> ↑v i * a_f i * e i x + ↑v i * a_g i * e i x
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
af_repr: f_repr v e (↑f) (Set.Nonempty.some ⋯)
right✝¹: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
ag_repr: f_repr v e (↑g) (Set.Nonempty.some ⋯)
right✝: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
summand_distrib: ∀ (x : ↑Ω) (i : ℕ), ↑v i * (a_f i + a_g i) * e i x = ↑v i * a_f i * e i x + ↑v i * a_g i * e i x
x: ↑Ω
intro.intro.right
Summable fun i => ↑v i * a_f i * e i x + ↑v i * a_g i * e i x]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
af_repr: f_repr v e (↑f) (Set.Nonempty.some ⋯)
right✝¹: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
ag_repr: f_repr v e (↑g) (Set.Nonempty.some ⋯)
right✝: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
summand_distrib: ∀ (x : ↑Ω) (i : ℕ), ↑v i * (a_f i + a_g i) * e i x = ↑v i * a_f i * e i x + ↑v i * a_g i * e i x
x: ↑Ω
intro.intro.right
Summable fun i => ↑v i * a_f i * e i x + ↑v i * a_g i * e i x
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
f_repr v e (fun x => ↑f x + ↑g x) fun i => Set.Nonempty.some ⋯ i + Set.Nonempty.some ⋯ iexact (af_repr.2 x).add (ag_repr.2 x)
Goals accomplished! 🐙

/-
 We define the 0 of H as pointwise 0 function. We show that it lies in H.
-/

def zero : Ω → ℝ := λ _ ↦ 0

lemma zero_repr : f_repr v e zero (λ _ ↦ 0) := by
Goals accomplished! 🐙
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
f_repr v e zero fun x => 0let a : ℕ → ℝ := λ _ ↦ 0
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= fun x => 0: ℕ → ℝ
f_repr v e zero fun x => 0
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
f_repr v e zero fun x => 0constructor
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= fun x => 0: ℕ → ℝ
left
zero = fun x => ∑' (i : ℕ), ↑v i * (fun x => 0) i * e i x
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= fun x => 0: ℕ → ℝ
right
∀ (x : ↑Ω), Summable fun i => ↑v i * (fun x => 0) i * e i x
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
f_repr v e zero fun x => 0·
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= fun x => 0: ℕ → ℝ
left
zero = fun x => ∑' (i : ℕ), ↑v i * (fun x => 0) i * e i x 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= fun x => 0: ℕ → ℝ
left
zero = fun x => ∑' (i : ℕ), ↑v i * (fun x => 0) i * e i x
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= fun x => 0: ℕ → ℝ
right
∀ (x : ↑Ω), Summable fun i => ↑v i * (fun x => 0) i * e i xext x
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= fun x => 0: ℕ → ℝ
x: ↑Ω
left.h
zero x = ∑' (i : ℕ), ↑v i * (fun x => 0) i * e i x
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= fun x => 0: ℕ → ℝ
left
zero = fun x => ∑' (i : ℕ), ↑v i * (fun x => 0) i * e i xhave summand_zero : ∀ i, v.1 i * a i * e i x = 0 := 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= fun x => 0: ℕ → ℝ
x: ↑Ω
left.h
zero x = ∑' (i : ℕ), ↑v i * (fun x => 0) i * e i xby
Goals accomplished! 🐙 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= fun x => 0: ℕ → ℝ
x: ↑Ω
∀ (i : ℕ), ↑v i * a i * e i x = 0{
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= fun x => 0: ℕ → ℝ
x: ↑Ω
∀ (i : ℕ), ↑v i * a i * e i x = 0
      
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= fun x => 0: ℕ → ℝ
x: ↑Ω
∀ (i : ℕ), ↑v i * a i * e i x = 0intro i
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= fun x => 0: ℕ → ℝ
x: ↑Ω
i: ℕ
↑v i * a i * e i x = 0
      
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= fun x => 0: ℕ → ℝ
x: ↑Ω
∀ (i : ℕ), ↑v i * a i * e i x = 0rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= fun x => 0: ℕ → ℝ
x: ↑Ω
i: ℕ
↑v i * a i * e i x = 0show v.1 i * a i * e i x = v.1 i * 0 * e i x by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= fun x => 0: ℕ → ℝ
x: ↑Ω
i: ℕ
↑v i * a i * e i x = 0rfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= fun x => 0: ℕ → ℝ
x: ↑Ω
i: ℕ
↑v i * 0 * e i x = 0
      
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= fun x => 0: ℕ → ℝ
x: ↑Ω
∀ (i : ℕ), ↑v i * a i * e i x = 0ring
Goals accomplished! 🐙
    }
Goals accomplished! 🐙
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= fun x => 0: ℕ → ℝ
left
zero = fun x => ∑' (i : ℕ), ↑v i * (fun x => 0) i * e i xsimp_rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= fun x => 0: ℕ → ℝ
x: ↑Ω
summand_zero: ∀ (i : ℕ), ↑v i * a i * e i x = 0
left.h
zero x = ∑' (i : ℕ), ↑v i * 0 * e i xsummand_zero
[Meta.Tactic.simp.rewrite] summand_zero:1000, ↑v i * 0 * e i x ==> 0
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= fun x => 0: ℕ → ℝ
x: ↑Ω
summand_zero: ∀ (i : ℕ), ↑v i * a i * e i x = 0
left.h
zero x = ∑' (i : ℕ), ↑v i * 0 * e i x,
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= fun x => 0: ℕ → ℝ
x: ↑Ω
summand_zero: ∀ (i : ℕ), ↑v i * a i * e i x = 0
left.h
zero x = ∑' (i : ℕ), 0 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= fun x => 0: ℕ → ℝ
x: ↑Ω
summand_zero: ∀ (i : ℕ), ↑v i * a i * e i x = 0
left.h
zero x = ∑' (i : ℕ), ↑v i * (fun x => 0) i * e i xtsum_zero
[Meta.Tactic.simp.rewrite] @tsum_zero:1000, ∑' (i : ℕ), 0 ==> 0
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= fun x => 0: ℕ → ℝ
x: ↑Ω
summand_zero: ∀ (i : ℕ), ↑v i * a i * e i x = 0
left.h
zero x = 0]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= fun x => 0: ℕ → ℝ
x: ↑Ω
summand_zero: ∀ (i : ℕ), ↑v i * a i * e i x = 0
left.h
zero x = 0
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= fun x => 0: ℕ → ℝ
left
zero = fun x => ∑' (i : ℕ), ↑v i * (fun x => 0) i * e i xrfl
Goals accomplished! 🐙
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
f_repr v e zero fun x => 0intro x
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= fun x => 0: ℕ → ℝ
x: ↑Ω
right
Summable fun i => ↑v i * (fun x => 0) i * e i x
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
f_repr v e zero fun x => 0have null_function : (λ i ↦ (v.1 i) * (a i) * (e i x)) = (λ (
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= fun x => 0: ℕ → ℝ
x: ↑Ω
right
Summable fun i => ↑v i * (fun x => 0) i * e i xi
Warning: unused variable `i` [linter.unusedVariables]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= fun x => 0: ℕ → ℝ
x: ↑Ω
right
Summable fun i => ↑v i * (fun x => 0) i * e i x : ℕ) ↦ (0 : ℝ)) := 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= fun x => 0: ℕ → ℝ
x: ↑Ω
right
Summable fun i => ↑v i * (fun x => 0) i * e i xby
Goals accomplished! 🐙 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= fun x => 0: ℕ → ℝ
x: ↑Ω
(fun i => ↑v i * a i * e i x) = fun i => 0{
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= fun x => 0: ℕ → ℝ
x: ↑Ω
(fun i => ↑v i * a i * e i x) = fun i => 0
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= fun x => 0: ℕ → ℝ
x: ↑Ω
(fun i => ↑v i * a i * e i x) = fun i => 0ext i
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= fun x => 0: ℕ → ℝ
x: ↑Ω
i: ℕ
h
↑v i * a i * e i x = 0
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= fun x => 0: ℕ → ℝ
x: ↑Ω
(fun i => ↑v i * a i * e i x) = fun i => 0rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= fun x => 0: ℕ → ℝ
x: ↑Ω
i: ℕ
h
↑v i * a i * e i x = 0show a i = 0 by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= fun x => 0: ℕ → ℝ
x: ↑Ω
i: ℕ
h
↑v i * a i * e i x = 0rfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= fun x => 0: ℕ → ℝ
x: ↑Ω
i: ℕ
h
↑v i * 0 * e i x = 0
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= fun x => 0: ℕ → ℝ
x: ↑Ω
(fun i => ↑v i * a i * e i x) = fun i => 0ring
Goals accomplished! 🐙
  }
Goals accomplished! 🐙
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
f_repr v e zero fun x => 0rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= fun x => 0: ℕ → ℝ
x: ↑Ω
null_function: (fun i => ↑v i * a i * e i x) = fun i => 0
right
Summable fun i => ↑v i * (fun x => 0) i * e i xnull_function
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= fun x => 0: ℕ → ℝ
x: ↑Ω
null_function: (fun i => ↑v i * a i * e i x) = fun i => 0
right
Summable fun i => 0]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= fun x => 0: ℕ → ℝ
x: ↑Ω
null_function: (fun i => ↑v i * a i * e i x) = fun i => 0
right
Summable fun i => 0
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
f_repr v e zero fun x => 0exact summable_zero
Goals accomplished! 🐙

lemma zero_summable : Summable (λ i ↦ (v.1 i) * (0 : ℝ)^2) := by
Goals accomplished! 🐙
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
Summable fun i => ↑v i * 0 ^ 2have zero_fun : (λ i ↦ v.1 i * (0 : ℝ)^2) = (λ (
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
Summable fun i => ↑v i * 0 ^ 2i
Warning: unused variable `i` [linter.unusedVariables]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
Summable fun i => ↑v i * 0 ^ 2 : ℕ) ↦ (0 : ℝ)) := 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
Summable fun i => ↑v i * 0 ^ 2by
Goals accomplished! 🐙 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
(fun i => ↑v i * 0 ^ 2) = fun i => 0{
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
(fun i => ↑v i * 0 ^ 2) = fun i => 0
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
(fun i => ↑v i * 0 ^ 2) = fun i => 0ext i
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
i: ℕ
h
↑v i * 0 ^ 2 = 0
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
(fun i => ↑v i * 0 ^ 2) = fun i => 0ring
Goals accomplished! 🐙
  }
Goals accomplished! 🐙
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
Summable fun i => ↑v i * 0 ^ 2rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
zero_fun: (fun i => ↑v i * 0 ^ 2) = fun i => 0
Summable fun i => ↑v i * 0 ^ 2zero_fun
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
zero_fun: (fun i => ↑v i * 0 ^ 2) = fun i => 0
Summable fun i => 0]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
zero_fun: (fun i => ↑v i * 0 ^ 2) = fun i => 0
Summable fun i => 0
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
Summable fun i => ↑v i * 0 ^ 2have hf : ∀ b ∉ (∅ : Finset ℕ), (λ (
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
zero_fun: (fun i => ↑v i * 0 ^ 2) = fun i => 0
Summable fun i => 0i
Warning: unused variable `i` [linter.unusedVariables]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
zero_fun: (fun i => ↑v i * 0 ^ 2) = fun i => 0
Summable fun i => 0 : ℕ) ↦ (0 : ℝ)) b = 0 := 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
zero_fun: (fun i => ↑v i * 0 ^ 2) = fun i => 0
Summable fun i => 0by
Goals accomplished! 🐙 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
zero_fun: (fun i => ↑v i * 0 ^ 2) = fun i => 0
∀ b ∉ ∅, (fun i => 0) b = 0{
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
zero_fun: (fun i => ↑v i * 0 ^ 2) = fun i => 0
∀ b ∉ ∅, (fun i => 0) b = 0
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
zero_fun: (fun i => ↑v i * 0 ^ 2) = fun i => 0
∀ b ∉ ∅, (fun i => 0) b = 0intro b 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
zero_fun: (fun i => ↑v i * 0 ^ 2) = fun i => 0
∀ b ∉ ∅, (fun i => 0) b = 0b_not_in
Warning: unused variable `b_not_in` [linter.unusedVariables]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
zero_fun: (fun i => ↑v i * 0 ^ 2) = fun i => 0
b: ℕ
b_not_in: b ∉ ∅
(fun i => 0) b = 0
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
zero_fun: (fun i => ↑v i * 0 ^ 2) = fun i => 0
∀ b ∉ ∅, (fun i => 0) b = 0rfl
Goals accomplished! 🐙
  }
Goals accomplished! 🐙
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
Summable fun i => ↑v i * 0 ^ 2exact summable_of_ne_finset_zero hf
Goals accomplished! 🐙

lemma zero_in_H : zero ∈ L2 μ ∧ ∃ (a : ℕ → ℝ), (f_repr v e zero a) ∧ Summable (λ i ↦ (v.1 i) * (a i)^2) := ⟨memℒp_const 0, (λ _ ↦ 0), zero_repr, zero_summable⟩

instance : Zero (H v e μ) where
  zero := ⟨zero, zero_in_H⟩

lemma zero_unique_repr : (set_repr_ne (0 : H v e μ)).some = (λ i ↦ 0) := by
Goals accomplished! 🐙
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
Set.Nonempty.some ⋯ = fun i => 0let a : ℕ → ℝ := λ _ ↦ 0
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= fun x => 0: ℕ → ℝ
Set.Nonempty.some ⋯ = fun i => 0
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
Set.Nonempty.some ⋯ = fun i => 0have a_in_repr : a ∈ set_repr (0 : H v e μ) := 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= fun x => 0: ℕ → ℝ
Set.Nonempty.some ⋯ = fun i => 0by
Goals accomplished! 🐙 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= fun x => 0: ℕ → ℝ
a ∈ set_repr 0{
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= fun x => 0: ℕ → ℝ
a ∈ set_repr 0
    --have tmp := zero_repr
    refine ⟨(zero_repr : f_repr v e (0 : H v e μ).1 a), ?_⟩
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= fun x => 0: ℕ → ℝ
Summable fun i => ↑v i * a i ^ 2
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= fun x => 0: ℕ → ℝ
a ∈ set_repr 0have null_function : (λ i ↦ v.1 i * (a i)^2) = λ (
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= fun x => 0: ℕ → ℝ
Summable fun i => ↑v i * a i ^ 2i
Warning: unused variable `i` [linter.unusedVariables]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= fun x => 0: ℕ → ℝ
Summable fun i => ↑v i * a i ^ 2 : ℕ) ↦ (0 : ℝ) := 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= fun x => 0: ℕ → ℝ
Summable fun i => ↑v i * a i ^ 2by
Goals accomplished! 🐙 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= fun x => 0: ℕ → ℝ
(fun i => ↑v i * a i ^ 2) = fun i => 0{
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= fun x => 0: ℕ → ℝ
(fun i => ↑v i * a i ^ 2) = fun i => 0
      
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= fun x => 0: ℕ → ℝ
(fun i => ↑v i * a i ^ 2) = fun i => 0ext i
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= fun x => 0: ℕ → ℝ
i: ℕ
h
↑v i * a i ^ 2 = 0
      
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= fun x => 0: ℕ → ℝ
(fun i => ↑v i * a i ^ 2) = fun i => 0rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= fun x => 0: ℕ → ℝ
i: ℕ
h
↑v i * a i ^ 2 = 0show a i = 0 by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= fun x => 0: ℕ → ℝ
i: ℕ
h
↑v i * a i ^ 2 = 0rfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= fun x => 0: ℕ → ℝ
i: ℕ
h
↑v i * 0 ^ 2 = 0
      
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= fun x => 0: ℕ → ℝ
(fun i => ↑v i * a i ^ 2) = fun i => 0ring
Goals accomplished! 🐙
    }
Goals accomplished! 🐙
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= fun x => 0: ℕ → ℝ
a ∈ set_repr 0rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= fun x => 0: ℕ → ℝ
null_function: (fun i => ↑v i * a i ^ 2) = fun i => 0
Summable fun i => ↑v i * a i ^ 2null_function
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= fun x => 0: ℕ → ℝ
null_function: (fun i => ↑v i * a i ^ 2) = fun i => 0
Summable fun i => 0]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= fun x => 0: ℕ → ℝ
null_function: (fun i => ↑v i * a i ^ 2) = fun i => 0
Summable fun i => 0
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= fun x => 0: ℕ → ℝ
a ∈ set_repr 0exact summable_zero
Goals accomplished! 🐙
  }
Goals accomplished! 🐙
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
Set.Nonempty.some ⋯ = fun i => 0exact unique_choice a_in_repr
Goals accomplished! 🐙

lemma add_in_H : (λ x ↦ f.1 x + g.1 x) ∈ H v e μ := by
Goals accomplished! 🐙
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
(fun x => ↑f x + ↑g x) ∈ H v e μlet h := λ i ↦ (set_repr_ne f).some i + (set_repr_ne g).some i
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
h:= fun i => Set.Nonempty.some ⋯ i + Set.Nonempty.some ⋯ i: ℕ → ℝ
(fun x => ↑f x + ↑g x) ∈ H v e μ
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
(fun x => ↑f x + ↑g x) ∈ H v e μexact ⟨add_in_L2 f g, h, add_repr f g, add_summable f g⟩
Goals accomplished! 🐙

instance : HAdd (H v e μ) (H v e μ) (H v e μ) where
  hAdd := λ f g ↦ ⟨(λ x ↦ f.1 x + g.1 x), add_in_H f g⟩

instance : HSub (H v e μ) (H v e μ) (H v e μ) where
  hSub := λ f g ↦ f + (-1 : ℝ) * g

instance : Neg (H v e μ) where
  neg := λ f ↦ (-1 : ℝ) * f

lemma H_add_assoc (a b c : H v e μ) : a + b + c = a + (b + c) := by
Goals accomplished! 🐙
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: ↑(H v e μ)
a + b + c = a + (b + c)ext x
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: ↑(H v e μ)
x: ↑Ω
a.h
↑(a + b + c) x = ↑(a + (b + c)) x
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: ↑(H v e μ)
a + b + c = a + (b + c)rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: ↑(H v e μ)
x: ↑Ω
a.h
↑(a + b + c) x = ↑(a + (b + c)) xshow (a + b + c).1 x = a.1 x + b.1 x + c.1 x by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: ↑(H v e μ)
x: ↑Ω
a.h
↑(a + b + c) x = ↑(a + (b + c)) xrfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: ↑(H v e μ)
x: ↑Ω
a.h
↑a x + ↑b x + ↑c x = ↑(a + (b + c)) x
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: ↑(H v e μ)
a + b + c = a + (b + c)rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: ↑(H v e μ)
x: ↑Ω
a.h
↑a x + ↑b x + ↑c x = ↑(a + (b + c)) xshow (a + (b + c)).1 x = a.1 x + (b.1 x + c.1 x) by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: ↑(H v e μ)
x: ↑Ω
a.h
↑a x + ↑b x + ↑c x = ↑(a + (b + c)) xrfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: ↑(H v e μ)
x: ↑Ω
a.h
↑a x + ↑b x + ↑c x = ↑a x + (↑b x + ↑c x)
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: ↑(H v e μ)
a + b + c = a + (b + c)ring
Goals accomplished! 🐙

lemma H_add_comm (a b : H v e μ) : a + b = b + a := by
Goals accomplished! 🐙
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
a + b = b + aext x
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
x: ↑Ω
a.h
↑(a + b) x = ↑(b + a) x
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
a + b = b + arw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
x: ↑Ω
a.h
↑(a + b) x = ↑(b + a) xshow (a + b).1 x = a.1 x + b.1 x by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
x: ↑Ω
a.h
↑(a + b) x = ↑(b + a) xrfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
x: ↑Ω
a.h
↑a x + ↑b x = ↑(b + a) x
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
a + b = b + arw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
x: ↑Ω
a.h
↑a x + ↑b x = ↑(b + a) xshow (b + a).1 x = b.1 x + a.1 x by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
x: ↑Ω
a.h
↑a x + ↑b x = ↑(b + a) xrfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
x: ↑Ω
a.h
↑a x + ↑b x = ↑b x + ↑a x
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
a + b = b + aring
Goals accomplished! 🐙

lemma H_add_zero (a : H v e μ) : a + 0 = a := by
Goals accomplished! 🐙
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
a + 0 = aext x
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
x: ↑Ω
a.h
↑(a + 0) x = ↑a x
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
a + 0 = arw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
x: ↑Ω
a.h
↑(a + 0) x = ↑a xshow (a + 0).1 x = a.1 x + (0 : H v e μ).1 x by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
x: ↑Ω
a.h
↑(a + 0) x = ↑a xrfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
x: ↑Ω
a.h
↑a x + ↑0 x = ↑a x
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
a + 0 = arw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
x: ↑Ω
a.h
↑a x + ↑0 x = ↑a xshow (0 : H v e μ).1 x = 0 by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
x: ↑Ω
a.h
↑a x + ↑0 x = ↑a xrfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
x: ↑Ω
a.h
↑a x + 0 = ↑a x
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
a + 0 = aring
Goals accomplished! 🐙

lemma H_zero_add (a : H v e μ) : 0 + a = a := by
Goals accomplished! 🐙
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
0 + a = arw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
0 + a = aH_add_comm 0 a
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
a + 0 = a]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
a + 0 = a
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
0 + a = aexact H_add_zero a
Goals accomplished! 🐙

lemma H_add_left_neg (a : H v e μ) : -a + a = 0 := by
Goals accomplished! 🐙
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
-a + a = 0rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
-a + a = 0show -a = (-1 : ℝ) * a by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
-a + a = 0rfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
-1 * a + a = 0
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
-a + a = 0ext x
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
x: ↑Ω
a.h
↑(-1 * a + a) x = ↑0 x
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
-a + a = 0rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
x: ↑Ω
a.h
↑(-1 * a + a) x = ↑0 xshow ((-1 : ℝ) * a + a).1 x = ((-1 : ℝ) * a).1 x + a.1 x by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
x: ↑Ω
a.h
↑(-1 * a + a) x = ↑0 xrfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
x: ↑Ω
a.h
↑(-1 * a) x + ↑a x = ↑0 x
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
-a + a = 0rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
x: ↑Ω
a.h
↑(-1 * a) x + ↑a x = ↑0 xshow ((-1 : ℝ) * a).1 x = (-1 : ℝ) * a.1 x by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
x: ↑Ω
a.h
↑(-1 * a) x + ↑a x = ↑0 xrfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
x: ↑Ω
a.h
-1 * ↑a x + ↑a x = ↑0 x
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
-a + a = 0rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
x: ↑Ω
a.h
-1 * ↑a x + ↑a x = ↑0 xshow (0 : H v e μ).1 x = 0 by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
x: ↑Ω
a.h
-1 * ↑a x + ↑a x = ↑0 xrfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
x: ↑Ω
a.h
-1 * ↑a x + ↑a x = 0
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
-a + a = 0ring
Goals accomplished! 🐙

end Group

/-
  We define a function : H × H → ℝ. The purpose of the following is to prove that H endowed with this function is a inner product space.
-/
noncomputable def H_inner {v : eigen} {e : ℕ → Ω → ℝ} {μ : Measure Ω} [IsFiniteMeasure μ] (f g : H v e μ) : ℝ := ∑' i, (v.1 i) * ((set_repr_ne f).some i) * ((set_repr_ne g).some i)

/-
 - We show properties on the inner product of H and the induced norm.
-/

namespace Inner

open Ring Group

variable {v : eigen} {e : ℕ → Ω → ℝ} {μ : Measure Ω} [IsFiniteMeasure μ] (f g : H v e μ)

noncomputable instance : Inner ℝ (H v e μ) where
  inner := H_inner

noncomputable instance : Norm (H v e μ) where
  norm := λ f ↦ Real.sqrt (inner f f)

noncomputable instance : Dist (H v e μ) where
  dist := λ f g ↦ norm (f - g)

lemma inner_mul_left (a : ℝ) : inner (a * f) g = a * inner f g := by
Goals accomplished! 🐙
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
⟪a * f, g⟫_ℝ = a * ⟪f, g⟫_ℝlet h := (set_repr_ne f).some
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
⟪a * f, g⟫_ℝ = a * ⟪f, g⟫_ℝ
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
⟪a * f, g⟫_ℝ = a * ⟪f, g⟫_ℝlet h_af := λ i ↦ a * h i
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
h_af:= fun i => a * h i: ℕ → ℝ
⟪a * f, g⟫_ℝ = a * ⟪f, g⟫_ℝ

  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
⟪a * f, g⟫_ℝ = a * ⟪f, g⟫_ℝhave h_af_in : h_af ∈ set_repr (a * f) := ⟨mul_repr f a, mul_summable f a⟩
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
h_af:= fun i => a * h i: ℕ → ℝ
h_af_in: h_af ∈ set_repr (a * f)
⟪a * f, g⟫_ℝ = a * ⟪f, g⟫_ℝ
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
⟪a * f, g⟫_ℝ = a * ⟪f, g⟫_ℝrw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
h_af:= fun i => a * h i: ℕ → ℝ
h_af_in: h_af ∈ set_repr (a * f)
⟪a * f, g⟫_ℝ = a * ⟪f, g⟫_ℝshow inner (a * f) g = H_inner (a * f) g by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
h_af:= fun i => a * h i: ℕ → ℝ
h_af_in: h_af ∈ set_repr (a * f)
⟪a * f, g⟫_ℝ = a * ⟪f, g⟫_ℝrfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
h_af:= fun i => a * h i: ℕ → ℝ
h_af_in: h_af ∈ set_repr (a * f)
H_inner (a * f) g = a * ⟪f, g⟫_ℝ
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
⟪a * f, g⟫_ℝ = a * ⟪f, g⟫_ℝunfold H_inner
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
h_af:= fun i => a * h i: ℕ → ℝ
h_af_in: h_af ∈ set_repr (a * f)
∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * Set.Nonempty.some ⋯ i = a * ⟪f, g⟫_ℝ
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
⟪a * f, g⟫_ℝ = a * ⟪f, g⟫_ℝrw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
h_af:= fun i => a * h i: ℕ → ℝ
h_af_in: h_af ∈ set_repr (a * f)
∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * Set.Nonempty.some ⋯ i = a * ⟪f, g⟫_ℝunique_choice h_af_in
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
h_af:= fun i => a * h i: ℕ → ℝ
h_af_in: h_af ∈ set_repr (a * f)
∑' (i : ℕ), ↑v i * h_af i * Set.Nonempty.some ⋯ i = a * ⟪f, g⟫_ℝ]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
h_af:= fun i => a * h i: ℕ → ℝ
h_af_in: h_af ∈ set_repr (a * f)
∑' (i : ℕ), ↑v i * h_af i * Set.Nonempty.some ⋯ i = a * ⟪f, g⟫_ℝ
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
⟪a * f, g⟫_ℝ = a * ⟪f, g⟫_ℝhave comm_summand : ∀ i, v.1 i * h_af i * (set_repr_ne g).some i = a * v.1 i * h i * (set_repr_ne g).some i := 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
h_af:= fun i => a * h i: ℕ → ℝ
h_af_in: h_af ∈ set_repr (a * f)
∑' (i : ℕ), ↑v i * h_af i * Set.Nonempty.some ⋯ i = a * ⟪f, g⟫_ℝby
Goals accomplished! 🐙 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
h_af:= fun i => a * h i: ℕ → ℝ
h_af_in: h_af ∈ set_repr (a * f)
∀ (i : ℕ), ↑v i * h_af i * Set.Nonempty.some ⋯ i = a * ↑v i * h i * Set.Nonempty.some ⋯ i{
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
h_af:= fun i => a * h i: ℕ → ℝ
h_af_in: h_af ∈ set_repr (a * f)
∀ (i : ℕ), ↑v i * h_af i * Set.Nonempty.some ⋯ i = a * ↑v i * h i * Set.Nonempty.some ⋯ i
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
h_af:= fun i => a * h i: ℕ → ℝ
h_af_in: h_af ∈ set_repr (a * f)
∀ (i : ℕ), ↑v i * h_af i * Set.Nonempty.some ⋯ i = a * ↑v i * h i * Set.Nonempty.some ⋯ iintro i
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
h_af:= fun i => a * h i: ℕ → ℝ
h_af_in: h_af ∈ set_repr (a * f)
i: ℕ
↑v i * h_af i * Set.Nonempty.some ⋯ i = a * ↑v i * h i * Set.Nonempty.some ⋯ i
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
h_af:= fun i => a * h i: ℕ → ℝ
h_af_in: h_af ∈ set_repr (a * f)
∀ (i : ℕ), ↑v i * h_af i * Set.Nonempty.some ⋯ i = a * ↑v i * h i * Set.Nonempty.some ⋯ iring
Goals accomplished! 🐙
  }
Goals accomplished! 🐙
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
⟪a * f, g⟫_ℝ = a * ⟪f, g⟫_ℝhave lambda_comm : ∀ i, a * v.1 i * h i * (set_repr_ne g).some i = a * (λ i ↦ v.1 i * h i * (set_repr_ne g).some i) i := 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
h_af:= fun i => a * h i: ℕ → ℝ
h_af_in: h_af ∈ set_repr (a * f)
comm_summand: ∀ (i : ℕ), ↑v i * h_af i * Set.Nonempty.some ⋯ i = a * ↑v i * h i * Set.Nonempty.some ⋯ i
∑' (i : ℕ), ↑v i * h_af i * Set.Nonempty.some ⋯ i = a * ⟪f, g⟫_ℝby
Goals accomplished! 🐙 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
h_af:= fun i => a * h i: ℕ → ℝ
h_af_in: h_af ∈ set_repr (a * f)
comm_summand: ∀ (i : ℕ), ↑v i * h_af i * Set.Nonempty.some ⋯ i = a * ↑v i * h i * Set.Nonempty.some ⋯ i
∀ (i : ℕ), a * ↑v i * h i * Set.Nonempty.some ⋯ i = a * (fun i => ↑v i * h i * Set.Nonempty.some ⋯ i) i{
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
h_af:= fun i => a * h i: ℕ → ℝ
h_af_in: h_af ∈ set_repr (a * f)
comm_summand: ∀ (i : ℕ), ↑v i * h_af i * Set.Nonempty.some ⋯ i = a * ↑v i * h i * Set.Nonempty.some ⋯ i
∀ (i : ℕ), a * ↑v i * h i * Set.Nonempty.some ⋯ i = a * (fun i => ↑v i * h i * Set.Nonempty.some ⋯ i) i
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
h_af:= fun i => a * h i: ℕ → ℝ
h_af_in: h_af ∈ set_repr (a * f)
comm_summand: ∀ (i : ℕ), ↑v i * h_af i * Set.Nonempty.some ⋯ i = a * ↑v i * h i * Set.Nonempty.some ⋯ i
∀ (i : ℕ), a * ↑v i * h i * Set.Nonempty.some ⋯ i = a * (fun i => ↑v i * h i * Set.Nonempty.some ⋯ i) iintro i
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
h_af:= fun i => a * h i: ℕ → ℝ
h_af_in: h_af ∈ set_repr (a * f)
comm_summand: ∀ (i : ℕ), ↑v i * h_af i * Set.Nonempty.some ⋯ i = a * ↑v i * h i * Set.Nonempty.some ⋯ i
i: ℕ
a * ↑v i * h i * Set.Nonempty.some ⋯ i = a * (fun i => ↑v i * h i * Set.Nonempty.some ⋯ i) i
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
h_af:= fun i => a * h i: ℕ → ℝ
h_af_in: h_af ∈ set_repr (a * f)
comm_summand: ∀ (i : ℕ), ↑v i * h_af i * Set.Nonempty.some ⋯ i = a * ↑v i * h i * Set.Nonempty.some ⋯ i
∀ (i : ℕ), a * ↑v i * h i * Set.Nonempty.some ⋯ i = a * (fun i => ↑v i * h i * Set.Nonempty.some ⋯ i) iring
Goals accomplished! 🐙
  }
Goals accomplished! 🐙
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
⟪a * f, g⟫_ℝ = a * ⟪f, g⟫_ℝsimp_rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
h_af:= fun i => a * h i: ℕ → ℝ
h_af_in: h_af ∈ set_repr (a * f)
comm_summand: ∀ (i : ℕ), ↑v i * h_af i * Set.Nonempty.some ⋯ i = a * ↑v i * h i * Set.Nonempty.some ⋯ i
lambda_comm: ∀ (i : ℕ), a * ↑v i * h i * Set.Nonempty.some ⋯ i = a * (fun i => ↑v i * h i * Set.Nonempty.some ⋯ i) i
∑' (i : ℕ), ↑v i * h_af i * Set.Nonempty.some ⋯ i = a * ⟪f, g⟫_ℝcomm_summand
[Meta.Tactic.simp.rewrite] comm_summand:1000, ↑v i * h_af i *
      Set.Nonempty.some ⋯ i ==> a * ↑v i * h i * Set.Nonempty.some ⋯ i
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
h_af:= fun i => a * h i: ℕ → ℝ
h_af_in: h_af ∈ set_repr (a * f)
comm_summand: ∀ (i : ℕ), ↑v i * h_af i * Set.Nonempty.some ⋯ i = a * ↑v i * h i * Set.Nonempty.some ⋯ i
lambda_comm: ∀ (i : ℕ), a * ↑v i * h i * Set.Nonempty.some ⋯ i = a * (fun i => ↑v i * h i * Set.Nonempty.some ⋯ i) i
∑' (i : ℕ), ↑v i * h_af i * Set.Nonempty.some ⋯ i = a * ⟪f, g⟫_ℝ,
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
h_af:= fun i => a * h i: ℕ → ℝ
h_af_in: h_af ∈ set_repr (a * f)
comm_summand: ∀ (i : ℕ), ↑v i * h_af i * Set.Nonempty.some ⋯ i = a * ↑v i * h i * Set.Nonempty.some ⋯ i
lambda_comm: ∀ (i : ℕ), a * ↑v i * h i * Set.Nonempty.some ⋯ i = a * (fun i => ↑v i * h i * Set.Nonempty.some ⋯ i) i
∑' (i : ℕ), a * ↑v i * h i * Set.Nonempty.some ⋯ i = a * ⟪f, g⟫_ℝ 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
h_af:= fun i => a * h i: ℕ → ℝ
h_af_in: h_af ∈ set_repr (a * f)
comm_summand: ∀ (i : ℕ), ↑v i * h_af i * Set.Nonempty.some ⋯ i = a * ↑v i * h i * Set.Nonempty.some ⋯ i
lambda_comm: ∀ (i : ℕ), a * ↑v i * h i * Set.Nonempty.some ⋯ i = a * (fun i => ↑v i * h i * Set.Nonempty.some ⋯ i) i
∑' (i : ℕ), ↑v i * h_af i * Set.Nonempty.some ⋯ i = a * ⟪f, g⟫_ℝlambda_comm
[Meta.Tactic.simp.rewrite] lambda_comm:1000, a * ↑v i * h i *
      Set.Nonempty.some ⋯ i ==> a * (fun i => ↑v i * h i * Set.Nonempty.some ⋯ i) i
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
h_af:= fun i => a * h i: ℕ → ℝ
h_af_in: h_af ∈ set_repr (a * f)
comm_summand: ∀ (i : ℕ), ↑v i * h_af i * Set.Nonempty.some ⋯ i = a * ↑v i * h i * Set.Nonempty.some ⋯ i
lambda_comm: ∀ (i : ℕ), a * ↑v i * h i * Set.Nonempty.some ⋯ i = a * (fun i => ↑v i * h i * Set.Nonempty.some ⋯ i) i
∑' (i : ℕ), a * (↑v i * h i * Set.Nonempty.some ⋯ i) = a * ⟪f, g⟫_ℝ]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
h:= Set.Nonempty.some ⋯: ℕ → ℝ
h_af:= fun i => a * h i: ℕ → ℝ
h_af_in: h_af ∈ set_repr (a * f)
comm_summand: ∀ (i : ℕ), ↑v i * h_af i * Set.Nonempty.some ⋯ i = a * ↑v i * h i * Set.Nonempty.some ⋯ i
lambda_comm: ∀ (i : ℕ), a * ↑v i * h i * Set.Nonempty.some ⋯ i = a * (fun i => ↑v i * h i * Set.Nonempty.some ⋯ i) i
∑' (i : ℕ), a * (↑v i * h i * Set.Nonempty.some ⋯ i) = a * ⟪f, g⟫_ℝ
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a: ℝ
⟪a * f, g⟫_ℝ = a * ⟪f, g⟫_ℝexact tsum_mul_left
Goals accomplished! 🐙

lemma H_inner_add_left (h : H v e μ) : (inner (f + g) h : ℝ) = inner f h + inner g h := by
Goals accomplished! 🐙
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, h: ↑(H v e μ)
⟪f + g, h⟫_ℝ = ⟪f, h⟫_ℝ + ⟪g, h⟫_ℝlet a_f := (set_repr_ne f).some
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, h: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
⟪f + g, h⟫_ℝ = ⟪f, h⟫_ℝ + ⟪g, h⟫_ℝ
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, h: ↑(H v e μ)
⟪f + g, h⟫_ℝ = ⟪f, h⟫_ℝ + ⟪g, h⟫_ℝlet a_g := (set_repr_ne g).some
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, h: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
⟪f + g, h⟫_ℝ = ⟪f, h⟫_ℝ + ⟪g, h⟫_ℝ
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, h: ↑(H v e μ)
⟪f + g, h⟫_ℝ = ⟪f, h⟫_ℝ + ⟪g, h⟫_ℝlet a_h := (set_repr_ne h).some
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, h: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_h:= Set.Nonempty.some ⋯: ℕ → ℝ
⟪f + g, h⟫_ℝ = ⟪f, h⟫_ℝ + ⟪g, h⟫_ℝ
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, h: ↑(H v e μ)
⟪f + g, h⟫_ℝ = ⟪f, h⟫_ℝ + ⟪g, h⟫_ℝlet a_fg := λ i ↦ a_f i + a_g i
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, h: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_h:= Set.Nonempty.some ⋯: ℕ → ℝ
a_fg:= fun i => a_f i + a_g i: ℕ → ℝ
⟪f + g, h⟫_ℝ = ⟪f, h⟫_ℝ + ⟪g, h⟫_ℝ

  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, h: ↑(H v e μ)
⟪f + g, h⟫_ℝ = ⟪f, h⟫_ℝ + ⟪g, h⟫_ℝhave a_fg_repr : a_fg ∈ set_repr (f + g) := ⟨add_repr f g, add_summable f g⟩
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, h: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_h:= Set.Nonempty.some ⋯: ℕ → ℝ
a_fg:= fun i => a_f i + a_g i: ℕ → ℝ
a_fg_repr: a_fg ∈ set_repr (f + g)
⟪f + g, h⟫_ℝ = ⟪f, h⟫_ℝ + ⟪g, h⟫_ℝ
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, h: ↑(H v e μ)
⟪f + g, h⟫_ℝ = ⟪f, h⟫_ℝ + ⟪g, h⟫_ℝrw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, h: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_h:= Set.Nonempty.some ⋯: ℕ → ℝ
a_fg:= fun i => a_f i + a_g i: ℕ → ℝ
a_fg_repr: a_fg ∈ set_repr (f + g)
⟪f + g, h⟫_ℝ = ⟪f, h⟫_ℝ + ⟪g, h⟫_ℝshow inner (f + g) h = H_inner (f + g) h by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, h: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_h:= Set.Nonempty.some ⋯: ℕ → ℝ
a_fg:= fun i => a_f i + a_g i: ℕ → ℝ
a_fg_repr: a_fg ∈ set_repr (f + g)
⟪f + g, h⟫_ℝ = ⟪f, h⟫_ℝ + ⟪g, h⟫_ℝrfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, h: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_h:= Set.Nonempty.some ⋯: ℕ → ℝ
a_fg:= fun i => a_f i + a_g i: ℕ → ℝ
a_fg_repr: a_fg ∈ set_repr (f + g)
H_inner (f + g) h = ⟪f, h⟫_ℝ + ⟪g, h⟫_ℝ
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, h: ↑(H v e μ)
⟪f + g, h⟫_ℝ = ⟪f, h⟫_ℝ + ⟪g, h⟫_ℝunfold H_inner
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, h: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_h:= Set.Nonempty.some ⋯: ℕ → ℝ
a_fg:= fun i => a_f i + a_g i: ℕ → ℝ
a_fg_repr: a_fg ∈ set_repr (f + g)
∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * Set.Nonempty.some ⋯ i = ⟪f, h⟫_ℝ + ⟪g, h⟫_ℝ
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, h: ↑(H v e μ)
⟪f + g, h⟫_ℝ = ⟪f, h⟫_ℝ + ⟪g, h⟫_ℝrw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, h: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_h:= Set.Nonempty.some ⋯: ℕ → ℝ
a_fg:= fun i => a_f i + a_g i: ℕ → ℝ
a_fg_repr: a_fg ∈ set_repr (f + g)
∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * Set.Nonempty.some ⋯ i = ⟪f, h⟫_ℝ + ⟪g, h⟫_ℝunique_choice a_fg_repr
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, h: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_h:= Set.Nonempty.some ⋯: ℕ → ℝ
a_fg:= fun i => a_f i + a_g i: ℕ → ℝ
a_fg_repr: a_fg ∈ set_repr (f + g)
∑' (i : ℕ), ↑v i * a_fg i * Set.Nonempty.some ⋯ i = ⟪f, h⟫_ℝ + ⟪g, h⟫_ℝ]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, h: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_h:= Set.Nonempty.some ⋯: ℕ → ℝ
a_fg:= fun i => a_f i + a_g i: ℕ → ℝ
a_fg_repr: a_fg ∈ set_repr (f + g)
∑' (i : ℕ), ↑v i * a_fg i * Set.Nonempty.some ⋯ i = ⟪f, h⟫_ℝ + ⟪g, h⟫_ℝ
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, h: ↑(H v e μ)
⟪f + g, h⟫_ℝ = ⟪f, h⟫_ℝ + ⟪g, h⟫_ℝsimp_rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, h: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_h:= Set.Nonempty.some ⋯: ℕ → ℝ
a_fg:= fun i => a_f i + a_g i: ℕ → ℝ
a_fg_repr: a_fg ∈ set_repr (f + g)
∑' (i : ℕ), ↑v i * a_fg i * Set.Nonempty.some ⋯ i = ⟪f, h⟫_ℝ + ⟪g, h⟫_ℝshow ∀ i, v.1 i * a_fg i * a_h i = v.1 i * (a_f i + a_g i) * a_h i by 
[Meta.Tactic.simp.rewrite] show ∀ i, v.1 i * a_fg i * a_h i = v.1 i * (a_f i + a_g i) * a_h i by intro i;
        rfl:1000, ↑v i * a_fg i * Set.Nonempty.some ⋯ i ==> ↑v i * (a_f i + a_g i) * a_h i
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, h: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_h:= Set.Nonempty.some ⋯: ℕ → ℝ
a_fg:= fun i => a_f i + a_g i: ℕ → ℝ
a_fg_repr: a_fg ∈ set_repr (f + g)
∑' (i : ℕ), ↑v i * a_fg i * Set.Nonempty.some ⋯ i = ⟪f, h⟫_ℝ + ⟪g, h⟫_ℝintro i
[Meta.Tactic.simp.rewrite] show ∀ i, v.1 i * a_fg i * a_h i = v.1 i * (a_f i + a_g i) * a_h i by intro i;
        rfl:1000, ↑v i * a_fg i * Set.Nonempty.some ⋯ i ==> ↑v i * (a_f i + a_g i) * a_h i
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, h: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_h:= Set.Nonempty.some ⋯: ℕ → ℝ
a_fg:= fun i => a_f i + a_g i: ℕ → ℝ
a_fg_repr: a_fg ∈ set_repr (f + g)
i: ℕ
↑v i * a_fg i * a_h i = ↑v i * (a_f i + a_g i) * a_h i;
[Meta.Tactic.simp.rewrite] show ∀ i, v.1 i * a_fg i * a_h i = v.1 i * (a_f i + a_g i) * a_h i by intro i;
        rfl:1000, ↑v i * a_fg i * Set.Nonempty.some ⋯ i ==> ↑v i * (a_f i + a_g i) * a_h i
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, h: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_h:= Set.Nonempty.some ⋯: ℕ → ℝ
a_fg:= fun i => a_f i + a_g i: ℕ → ℝ
a_fg_repr: a_fg ∈ set_repr (f + g)
i: ℕ
↑v i * a_fg i * a_h i = ↑v i * (a_f i + a_g i) * a_h i 
[Meta.Tactic.simp.rewrite] show ∀ i, v.1 i * a_fg i * a_h i = v.1 i * (a_f i + a_g i) * a_h i by intro i;
        rfl:1000, ↑v i * a_fg i * Set.Nonempty.some ⋯ i ==> ↑v i * (a_f i + a_g i) * a_h i
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, h: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_h:= Set.Nonempty.some ⋯: ℕ → ℝ
a_fg:= fun i => a_f i + a_g i: ℕ → ℝ
a_fg_repr: a_fg ∈ set_repr (f + g)
∀ (i : ℕ), ↑v i * a_fg i * a_h i = ↑v i * (a_f i + a_g i) * a_h irfl
[Meta.Tactic.simp.rewrite] show ∀ i, v.1 i * a_fg i * a_h i = v.1 i * (a_f i + a_g i) * a_h i by intro i;
        rfl:1000, ↑v i * a_fg i * Set.Nonempty.some ⋯ i ==> ↑v i * (a_f i + a_g i) * a_h i
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, h: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_h:= Set.Nonempty.some ⋯: ℕ → ℝ
a_fg:= fun i => a_f i + a_g i: ℕ → ℝ
a_fg_repr: a_fg ∈ set_repr (f + g)
∑' (i : ℕ), ↑v i * (a_f i + a_g i) * a_h i = ⟪f, h⟫_ℝ + ⟪g, h⟫_ℝ

  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, h: ↑(H v e μ)
⟪f + g, h⟫_ℝ = ⟪f, h⟫_ℝ + ⟪g, h⟫_ℝsimp_rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, h: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_h:= Set.Nonempty.some ⋯: ℕ → ℝ
a_fg:= fun i => a_f i + a_g i: ℕ → ℝ
a_fg_repr: a_fg ∈ set_repr (f + g)
∑' (i : ℕ), ↑v i * (a_f i + a_g i) * a_h i = ⟪f, h⟫_ℝ + ⟪g, h⟫_ℝshow ∀ i, v.1 i * (a_f i + a_g i) * a_h i = v.1 i * a_f i * a_h i + v.1 i * a_g i * a_h i by 
[Meta.Tactic.simp.rewrite] show ∀ i, v.1 i * (a_f i + a_g i) * a_h i = v.1 i * a_f i * a_h i + v.1 i * a_g i * a_h i by
        intro i; ring:1000, ↑v i * (a_f i + a_g i) * a_h i ==> ↑v i * a_f i * a_h i + ↑v i * a_g i * a_h i
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, h: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_h:= Set.Nonempty.some ⋯: ℕ → ℝ
a_fg:= fun i => a_f i + a_g i: ℕ → ℝ
a_fg_repr: a_fg ∈ set_repr (f + g)
∑' (i : ℕ), ↑v i * (a_f i + a_g i) * a_h i = ⟪f, h⟫_ℝ + ⟪g, h⟫_ℝintro i
[Meta.Tactic.simp.rewrite] show ∀ i, v.1 i * (a_f i + a_g i) * a_h i = v.1 i * a_f i * a_h i + v.1 i * a_g i * a_h i by
        intro i; ring:1000, ↑v i * (a_f i + a_g i) * a_h i ==> ↑v i * a_f i * a_h i + ↑v i * a_g i * a_h i
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, h: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_h:= Set.Nonempty.some ⋯: ℕ → ℝ
a_fg:= fun i => a_f i + a_g i: ℕ → ℝ
a_fg_repr: a_fg ∈ set_repr (f + g)
i: ℕ
↑v i * (a_f i + a_g i) * a_h i = ↑v i * a_f i * a_h i + ↑v i * a_g i * a_h i;
[Meta.Tactic.simp.rewrite] show ∀ i, v.1 i * (a_f i + a_g i) * a_h i = v.1 i * a_f i * a_h i + v.1 i * a_g i * a_h i by
        intro i; ring:1000, ↑v i * (a_f i + a_g i) * a_h i ==> ↑v i * a_f i * a_h i + ↑v i * a_g i * a_h i
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, h: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_h:= Set.Nonempty.some ⋯: ℕ → ℝ
a_fg:= fun i => a_f i + a_g i: ℕ → ℝ
a_fg_repr: a_fg ∈ set_repr (f + g)
i: ℕ
↑v i * (a_f i + a_g i) * a_h i = ↑v i * a_f i * a_h i + ↑v i * a_g i * a_h i 
[Meta.Tactic.simp.rewrite] show ∀ i, v.1 i * (a_f i + a_g i) * a_h i = v.1 i * a_f i * a_h i + v.1 i * a_g i * a_h i by
        intro i; ring:1000, ↑v i * (a_f i + a_g i) * a_h i ==> ↑v i * a_f i * a_h i + ↑v i * a_g i * a_h i
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, h: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_h:= Set.Nonempty.some ⋯: ℕ → ℝ
a_fg:= fun i => a_f i + a_g i: ℕ → ℝ
a_fg_repr: a_fg ∈ set_repr (f + g)
∀ (i : ℕ), ↑v i * (a_f i + a_g i) * a_h i = ↑v i * a_f i * a_h i + ↑v i * a_g i * a_h iring
[Meta.Tactic.simp.rewrite] show ∀ i, v.1 i * (a_f i + a_g i) * a_h i = v.1 i * a_f i * a_h i + v.1 i * a_g i * a_h i by
        intro i; ring:1000, ↑v i * (a_f i + a_g i) * a_h i ==> ↑v i * a_f i * a_h i + ↑v i * a_g i * a_h i
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, h: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_h:= Set.Nonempty.some ⋯: ℕ → ℝ
a_fg:= fun i => a_f i + a_g i: ℕ → ℝ
a_fg_repr: a_fg ∈ set_repr (f + g)
∑' (i : ℕ), (↑v i * a_f i * a_h i + ↑v i * a_g i * a_h i) = ⟪f, h⟫_ℝ + ⟪g, h⟫_ℝ

  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, h: ↑(H v e μ)
⟪f + g, h⟫_ℝ = ⟪f, h⟫_ℝ + ⟪g, h⟫_ℝrw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, h: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_h:= Set.Nonempty.some ⋯: ℕ → ℝ
a_fg:= fun i => a_f i + a_g i: ℕ → ℝ
a_fg_repr: a_fg ∈ set_repr (f + g)
∑' (i : ℕ), (↑v i * a_f i * a_h i + ↑v i * a_g i * a_h i) = ⟪f, h⟫_ℝ + ⟪g, h⟫_ℝtsum_add (product_summable f h) (product_summable g h)
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, h: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_h:= Set.Nonempty.some ⋯: ℕ → ℝ
a_fg:= fun i => a_f i + a_g i: ℕ → ℝ
a_fg_repr: a_fg ∈ set_repr (f + g)
∑' (b : ℕ), ↑v b * Set.Nonempty.some ⋯ b * Set.Nonempty.some ⋯ b +
    ∑' (b : ℕ), ↑v b * Set.Nonempty.some ⋯ b * Set.Nonempty.some ⋯ b =
  ⟪f, h⟫_ℝ + ⟪g, h⟫_ℝ]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, h: ↑(H v e μ)
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_h:= Set.Nonempty.some ⋯: ℕ → ℝ
a_fg:= fun i => a_f i + a_g i: ℕ → ℝ
a_fg_repr: a_fg ∈ set_repr (f + g)
∑' (b : ℕ), ↑v b * Set.Nonempty.some ⋯ b * Set.Nonempty.some ⋯ b +
    ∑' (b : ℕ), ↑v b * Set.Nonempty.some ⋯ b * Set.Nonempty.some ⋯ b =
  ⟪f, h⟫_ℝ + ⟪g, h⟫_ℝ
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, h: ↑(H v e μ)
⟪f + g, h⟫_ℝ = ⟪f, h⟫_ℝ + ⟪g, h⟫_ℝrfl
Goals accomplished! 🐙

lemma inner_symmetric : (inner f g : ℝ) = inner g f := by
Goals accomplished! 🐙
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
⟪f, g⟫_ℝ = ⟪g, f⟫_ℝrw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
⟪f, g⟫_ℝ = ⟪g, f⟫_ℝshow inner f g = H_inner f g by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
⟪f, g⟫_ℝ = ⟪g, f⟫_ℝrfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
H_inner f g = ⟪g, f⟫_ℝ
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
⟪f, g⟫_ℝ = ⟪g, f⟫_ℝunfold H_inner
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * Set.Nonempty.some ⋯ i = ⟪g, f⟫_ℝ
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
⟪f, g⟫_ℝ = ⟪g, f⟫_ℝhave comm : ∀ i, (v.1 i) * ((set_repr_ne f).some i) * ((set_repr_ne g).some i) = (v.1 i) * ((set_repr_ne g).some i) * ((set_repr_ne f).some i) := λ i ↦ 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * Set.Nonempty.some ⋯ i = ⟪g, f⟫_ℝby
Goals accomplished! 🐙 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
i: ℕ
↑v i * Set.Nonempty.some ⋯ i * Set.Nonempty.some ⋯ i = ↑v i * Set.Nonempty.some ⋯ i * Set.Nonempty.some ⋯ iring
Goals accomplished! 🐙
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
⟪f, g⟫_ℝ = ⟪g, f⟫_ℝsimp_rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
comm: ∀ (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * Set.Nonempty.some ⋯ i = ↑v i * Set.Nonempty.some ⋯ i * Set.Nonempty.some ⋯ i
∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * Set.Nonempty.some ⋯ i = ⟪g, f⟫_ℝcomm
[Meta.Tactic.simp.rewrite] comm:1000, ↑v i * Set.Nonempty.some ⋯ i *
      Set.Nonempty.some ⋯ i ==> ↑v i * Set.Nonempty.some ⋯ i * Set.Nonempty.some ⋯ i
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
comm: ∀ (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * Set.Nonempty.some ⋯ i = ↑v i * Set.Nonempty.some ⋯ i * Set.Nonempty.some ⋯ i
∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * Set.Nonempty.some ⋯ i = ⟪g, f⟫_ℝ]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
comm: ∀ (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * Set.Nonempty.some ⋯ i = ↑v i * Set.Nonempty.some ⋯ i * Set.Nonempty.some ⋯ i
∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * Set.Nonempty.some ⋯ i = ⟪g, f⟫_ℝ
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
⟪f, g⟫_ℝ = ⟪g, f⟫_ℝrfl
Goals accomplished! 🐙

lemma inner_nonneg : (0 : ℝ) <= inner f f := by
Goals accomplished! 🐙
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
0 ≤ ⟪f, f⟫_ℝrw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
0 ≤ ⟪f, f⟫_ℝshow inner f f = H_inner f f by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
0 ≤ ⟪f, f⟫_ℝrfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
0 ≤ H_inner f f
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
0 ≤ ⟪f, f⟫_ℝunfold H_inner
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
0 ≤ ∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * Set.Nonempty.some ⋯ i
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
0 ≤ ⟪f, f⟫_ℝlet a := (set_repr_ne f).some
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
0 ≤ ∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * Set.Nonempty.some ⋯ i
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
0 ≤ ⟪f, f⟫_ℝhave sq : ∀ i, v.1 i * a i * a i = (v.1 i) * (a i)^2 := 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
0 ≤ ∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * Set.Nonempty.some ⋯ iby
Goals accomplished! 🐙 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2{
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2intro i
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
i: ℕ
↑v i * a i * a i = ↑v i * a i ^ 2
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2ring
Goals accomplished! 🐙
  }
Goals accomplished! 🐙
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
0 ≤ ⟪f, f⟫_ℝsimp_rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
0 ≤ ∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * Set.Nonempty.some ⋯ isq
[Meta.Tactic.simp.rewrite] sq:1000, ↑v i * Set.Nonempty.some ⋯ i * Set.Nonempty.some ⋯ i ==> ↑v i * a i ^ 2
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
0 ≤ ∑' (i : ℕ), ↑v i * a i ^ 2]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
0 ≤ ∑' (i : ℕ), ↑v i * a i ^ 2
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
0 ≤ ⟪f, f⟫_ℝhave nonneg : ∀ i, (0 : ℝ) <= (v.1 i) * (a i)^2 := 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
0 ≤ ∑' (i : ℕ), ↑v i * a i ^ 2by
Goals accomplished! 🐙 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
∀ (i : ℕ), 0 ≤ ↑v i * a i ^ 2{
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
∀ (i : ℕ), 0 ≤ ↑v i * a i ^ 2
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
∀ (i : ℕ), 0 ≤ ↑v i * a i ^ 2intro i
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
i: ℕ
0 ≤ ↑v i * a i ^ 2
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
∀ (i : ℕ), 0 ≤ ↑v i * a i ^ 2exact Left.mul_nonneg (v.2 i) (sq_nonneg (a i))
Goals accomplished! 🐙
  }
Goals accomplished! 🐙
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
0 ≤ ⟪f, f⟫_ℝexact tsum_nonneg nonneg
Goals accomplished! 🐙

lemma H_norm_nonneg : (0 : ℝ) <= norm f := by
Goals accomplished! 🐙
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
0 ≤ ‖f‖rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
0 ≤ ‖f‖show norm f = Real.sqrt (inner f f) by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
0 ≤ ‖f‖rfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
0 ≤ √⟪f, f⟫_ℝ
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
0 ≤ ‖f‖exact Real.sqrt_nonneg (inner f f)
Goals accomplished! 🐙

lemma inner_zero_eq_zero : inner f 0 = (0 : ℝ) := by
Goals accomplished! 🐙
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
⟪f, 0⟫_ℝ = 0rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
⟪f, 0⟫_ℝ = 0show inner f 0 = H_inner f 0 by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
⟪f, 0⟫_ℝ = 0rfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
H_inner f 0 = 0
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
⟪f, 0⟫_ℝ = 0unfold H_inner
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * Set.Nonempty.some ⋯ i = 0
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
⟪f, 0⟫_ℝ = 0rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * Set.Nonempty.some ⋯ i = 0zero_unique_repr
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * (fun i => 0) i = 0]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * (fun i => 0) i = 0

  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
⟪f, 0⟫_ℝ = 0have summand_eq_zero : ∀ i, v.1 i * (set_repr_ne f).some i * (λ i ↦ 0) i = 0 := 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * (fun i => 0) i = 0by
Goals accomplished! 🐙 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
∀ (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * (fun i => 0) i = 0{
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
∀ (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * (fun i => 0) i = 0
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
∀ (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * (fun i => 0) i = 0intro i
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
i: ℕ
↑v i * Set.Nonempty.some ⋯ i * (fun i => 0) i = 0
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
∀ (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * (fun i => 0) i = 0rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
i: ℕ
↑v i * Set.Nonempty.some ⋯ i * (fun i => 0) i = 0show (λ (
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
i: ℕ
↑v i * Set.Nonempty.some ⋯ i * (fun i => 0) i = 0i
Warning: unused variable `i` [linter.unusedVariables]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
i: ℕ
↑v i * Set.Nonempty.some ⋯ i * (fun i => 0) i = 0 : ℕ) ↦ (0 : ℝ)) i = 0 by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
i: ℕ
↑v i * Set.Nonempty.some ⋯ i * (fun i => 0) i = 0rfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
i: ℕ
↑v i * Set.Nonempty.some ⋯ i * 0 = 0
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
∀ (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * (fun i => 0) i = 0ring
Goals accomplished! 🐙
  }
Goals accomplished! 🐙
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
⟪f, 0⟫_ℝ = 0simp_rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
summand_eq_zero: ∀ (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * (fun i => 0) i = 0
∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * 0 = 0summand_eq_zero
[Meta.Tactic.simp.rewrite] summand_eq_zero:1000, ↑v i * Set.Nonempty.some ⋯ i * 0 ==> 0
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
summand_eq_zero: ∀ (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * (fun i => 0) i = 0
∑' (i : ℕ), 0 = 0]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
summand_eq_zero: ∀ (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * (fun i => 0) i = 0
∑' (i : ℕ), 0 = 0
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
⟪f, 0⟫_ℝ = 0exact tsum_zero
Goals accomplished! 🐙

lemma null_inner_imp_null_f : inner f f = (0 : ℝ) → f = 0 := by
Goals accomplished! 🐙
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
⟪f, f⟫_ℝ = 0 → f = 0intro inner_eq_0
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
inner_eq_0: ⟪f, f⟫_ℝ = 0
f = 0
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
⟪f, f⟫_ℝ = 0 → f = 0rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
inner_eq_0: ⟪f, f⟫_ℝ = 0
f = 0show inner f f = H_inner f f by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
inner_eq_0: ⟪f, f⟫_ℝ = 0
f = 0rfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
inner_eq_0: H_inner f f = 0
f = 0 at inner_eq_0
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
inner_eq_0: H_inner f f = 0
f = 0
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
⟪f, f⟫_ℝ = 0 → f = 0unfold H_inner at inner_eq_0
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
inner_eq_0: ∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * Set.Nonempty.some ⋯ i = 0
f = 0

  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
⟪f, f⟫_ℝ = 0 → f = 0let a := (set_repr_ne f).some
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
inner_eq_0: ∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * Set.Nonempty.some ⋯ i = 0
a:= Set.Nonempty.some ⋯: ℕ → ℝ
f = 0

  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
⟪f, f⟫_ℝ = 0 → f = 0have sq_summand : ∀ i, v.1 i * a i * a i = v.1 i * (a i)^2 := λ i ↦ 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
inner_eq_0: ∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * Set.Nonempty.some ⋯ i = 0
a:= Set.Nonempty.some ⋯: ℕ → ℝ
f = 0by
Goals accomplished! 🐙 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
inner_eq_0: ∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * Set.Nonempty.some ⋯ i = 0
a:= Set.Nonempty.some ⋯: ℕ → ℝ
i: ℕ
↑v i * a i * a i = ↑v i * a i ^ 2ring
Goals accomplished! 🐙
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
⟪f, f⟫_ℝ = 0 → f = 0simp_rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
inner_eq_0: ∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * Set.Nonempty.some ⋯ i = 0
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq_summand: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
f = 0sq_summand
[Meta.Tactic.simp.rewrite] sq_summand:1000, ↑v i * Set.Nonempty.some ⋯ i * Set.Nonempty.some ⋯ i ==> ↑v i * a i ^ 2
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq_summand: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), ↑v i * a i ^ 2 = 0
f = 0] at inner_eq_0
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq_summand: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), ↑v i * a i ^ 2 = 0
f = 0

  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
⟪f, f⟫_ℝ = 0 → f = 0have summand_nonneg : ∀ i, (0 : ℝ) <= v.1 i * (a i)^2 := 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq_summand: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), ↑v i * a i ^ 2 = 0
f = 0by
Goals accomplished! 🐙 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq_summand: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), ↑v i * a i ^ 2 = 0
∀ (i : ℕ), 0 ≤ ↑v i * a i ^ 2{
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq_summand: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), ↑v i * a i ^ 2 = 0
∀ (i : ℕ), 0 ≤ ↑v i * a i ^ 2
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq_summand: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), ↑v i * a i ^ 2 = 0
∀ (i : ℕ), 0 ≤ ↑v i * a i ^ 2intro i
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq_summand: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), ↑v i * a i ^ 2 = 0
i: ℕ
0 ≤ ↑v i * a i ^ 2
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq_summand: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), ↑v i * a i ^ 2 = 0
∀ (i : ℕ), 0 ≤ ↑v i * a i ^ 2exact Left.mul_nonneg (v.2 i) (sq_nonneg (a i))
Goals accomplished! 🐙
  }
Goals accomplished! 🐙

  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
⟪f, f⟫_ℝ = 0 → f = 0have summand_summable : Summable (λ i ↦ v.1 i * (a i)^2) := 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq_summand: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), ↑v i * a i ^ 2 = 0
summand_nonneg: ∀ (i : ℕ), 0 ≤ ↑v i * a i ^ 2
f = 0by
Goals accomplished! 🐙 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq_summand: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), ↑v i * a i ^ 2 = 0
summand_nonneg: ∀ (i : ℕ), 0 ≤ ↑v i * a i ^ 2
Summable fun i => ↑v i * a i ^ 2{
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq_summand: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), ↑v i * a i ^ 2 = 0
summand_nonneg: ∀ (i : ℕ), 0 ≤ ↑v i * a i ^ 2
Summable fun i => ↑v i * a i ^ 2
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq_summand: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), ↑v i * a i ^ 2 = 0
summand_nonneg: ∀ (i : ℕ), 0 ≤ ↑v i * a i ^ 2
Summable fun i => ↑v i * a i ^ 2simp_rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq_summand: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), ↑v i * a i ^ 2 = 0
summand_nonneg: ∀ (i : ℕ), 0 ≤ ↑v i * a i ^ 2
Summable fun i => ↑v i * a i ^ 2←sq_summand
[Meta.Tactic.simp.rewrite] ← sq_summand:1000, ↑v i * a i ^ 2 ==> ↑v i * a i * a i
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq_summand: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), ↑v i * a i ^ 2 = 0
summand_nonneg: ∀ (i : ℕ), 0 ≤ ↑v i * a i ^ 2
Summable fun i => ↑v i * a i * a i]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq_summand: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), ↑v i * a i ^ 2 = 0
summand_nonneg: ∀ (i : ℕ), 0 ≤ ↑v i * a i ^ 2
Summable fun i => ↑v i * a i * a i
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq_summand: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), ↑v i * a i ^ 2 = 0
summand_nonneg: ∀ (i : ℕ), 0 ≤ ↑v i * a i ^ 2
Summable fun i => ↑v i * a i ^ 2exact product_summable f f
Goals accomplished! 🐙
  }
Goals accomplished! 🐙

  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
⟪f, f⟫_ℝ = 0 → f = 0have summand_eq_zero := (summable_nonneg_iff_0 summand_nonneg summand_summable).mp inner_eq_0
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq_summand: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), ↑v i * a i ^ 2 = 0
summand_nonneg: ∀ (i : ℕ), 0 ≤ ↑v i * a i ^ 2
summand_summable: Summable fun i => ↑v i * a i ^ 2
summand_eq_zero: ∀ (i : ℕ), ↑v i * a i ^ 2 = 0
f = 0

  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
⟪f, f⟫_ℝ = 0 → f = 0have mul_v_a_eq_0 : ∀ i, v.1 i * a i = 0 := 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq_summand: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), ↑v i * a i ^ 2 = 0
summand_nonneg: ∀ (i : ℕ), 0 ≤ ↑v i * a i ^ 2
summand_summable: Summable fun i => ↑v i * a i ^ 2
summand_eq_zero: ∀ (i : ℕ), ↑v i * a i ^ 2 = 0
f = 0by
Goals accomplished! 🐙 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq_summand: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), ↑v i * a i ^ 2 = 0
summand_nonneg: ∀ (i : ℕ), 0 ≤ ↑v i * a i ^ 2
summand_summable: Summable fun i => ↑v i * a i ^ 2
summand_eq_zero: ∀ (i : ℕ), ↑v i * a i ^ 2 = 0
∀ (i : ℕ), ↑v i * a i = 0{
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq_summand: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), ↑v i * a i ^ 2 = 0
summand_nonneg: ∀ (i : ℕ), 0 ≤ ↑v i * a i ^ 2
summand_summable: Summable fun i => ↑v i * a i ^ 2
summand_eq_zero: ∀ (i : ℕ), ↑v i * a i ^ 2 = 0
∀ (i : ℕ), ↑v i * a i = 0
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq_summand: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), ↑v i * a i ^ 2 = 0
summand_nonneg: ∀ (i : ℕ), 0 ≤ ↑v i * a i ^ 2
summand_summable: Summable fun i => ↑v i * a i ^ 2
summand_eq_zero: ∀ (i : ℕ), ↑v i * a i ^ 2 = 0
∀ (i : ℕ), ↑v i * a i = 0intro i
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq_summand: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), ↑v i * a i ^ 2 = 0
summand_nonneg: ∀ (i : ℕ), 0 ≤ ↑v i * a i ^ 2
summand_summable: Summable fun i => ↑v i * a i ^ 2
summand_eq_zero: ∀ (i : ℕ), ↑v i * a i ^ 2 = 0
i: ℕ
↑v i * a i = 0
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq_summand: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), ↑v i * a i ^ 2 = 0
summand_nonneg: ∀ (i : ℕ), 0 ≤ ↑v i * a i ^ 2
summand_summable: Summable fun i => ↑v i * a i ^ 2
summand_eq_zero: ∀ (i : ℕ), ↑v i * a i ^ 2 = 0
∀ (i : ℕ), ↑v i * a i = 0cases mul_eq_zero.mp (summand_eq_zero i) with
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq_summand: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), ↑v i * a i ^ 2 = 0
summand_nonneg: ∀ (i : ℕ), 0 ≤ ↑v i * a i ^ 2
summand_summable: Summable fun i => ↑v i * a i ^ 2
summand_eq_zero: ∀ (i : ℕ), ↑v i * a i ^ 2 = 0
i: ℕ
x✝: ↑v i = 0 ∨ a i ^ 2 = 0
↑v i * a i = 0| inl hv =>
      
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq_summand: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), ↑v i * a i ^ 2 = 0
summand_nonneg: ∀ (i : ℕ), 0 ≤ ↑v i * a i ^ 2
summand_summable: Summable fun i => ↑v i * a i ^ 2
summand_eq_zero: ∀ (i : ℕ), ↑v i * a i ^ 2 = 0
i: ℕ
hv: ↑v i = 0
inl
↑v i * a i = 0rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq_summand: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), ↑v i * a i ^ 2 = 0
summand_nonneg: ∀ (i : ℕ), 0 ≤ ↑v i * a i ^ 2
summand_summable: Summable fun i => ↑v i * a i ^ 2
summand_eq_zero: ∀ (i : ℕ), ↑v i * a i ^ 2 = 0
i: ℕ
hv: ↑v i = 0
inl
↑v i * a i = 0hv
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq_summand: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), ↑v i * a i ^ 2 = 0
summand_nonneg: ∀ (i : ℕ), 0 ≤ ↑v i * a i ^ 2
summand_summable: Summable fun i => ↑v i * a i ^ 2
summand_eq_zero: ∀ (i : ℕ), ↑v i * a i ^ 2 = 0
i: ℕ
hv: ↑v i = 0
inl
0 * a i = 0]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq_summand: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), ↑v i * a i ^ 2 = 0
summand_nonneg: ∀ (i : ℕ), 0 ≤ ↑v i * a i ^ 2
summand_summable: Summable fun i => ↑v i * a i ^ 2
summand_eq_zero: ∀ (i : ℕ), ↑v i * a i ^ 2 = 0
i: ℕ
hv: ↑v i = 0
inl
0 * a i = 0
      
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq_summand: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), ↑v i * a i ^ 2 = 0
summand_nonneg: ∀ (i : ℕ), 0 ≤ ↑v i * a i ^ 2
summand_summable: Summable fun i => ↑v i * a i ^ 2
summand_eq_zero: ∀ (i : ℕ), ↑v i * a i ^ 2 = 0
i: ℕ
hv: ↑v i = 0
inl
↑v i * a i = 0ring
Goals accomplished! 🐙
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq_summand: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), ↑v i * a i ^ 2 = 0
summand_nonneg: ∀ (i : ℕ), 0 ≤ ↑v i * a i ^ 2
summand_summable: Summable fun i => ↑v i * a i ^ 2
summand_eq_zero: ∀ (i : ℕ), ↑v i * a i ^ 2 = 0
i: ℕ
x✝: ↑v i = 0 ∨ a i ^ 2 = 0
↑v i * a i = 0| inr ha =>
      
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq_summand: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), ↑v i * a i ^ 2 = 0
summand_nonneg: ∀ (i : ℕ), 0 ≤ ↑v i * a i ^ 2
summand_summable: Summable fun i => ↑v i * a i ^ 2
summand_eq_zero: ∀ (i : ℕ), ↑v i * a i ^ 2 = 0
i: ℕ
ha: a i ^ 2 = 0
inr
↑v i * a i = 0rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq_summand: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), ↑v i * a i ^ 2 = 0
summand_nonneg: ∀ (i : ℕ), 0 ≤ ↑v i * a i ^ 2
summand_summable: Summable fun i => ↑v i * a i ^ 2
summand_eq_zero: ∀ (i : ℕ), ↑v i * a i ^ 2 = 0
i: ℕ
ha: a i ^ 2 = 0
inr
↑v i * a i = 0sq_eq_zero_iff.mp ha
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq_summand: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), ↑v i * a i ^ 2 = 0
summand_nonneg: ∀ (i : ℕ), 0 ≤ ↑v i * a i ^ 2
summand_summable: Summable fun i => ↑v i * a i ^ 2
summand_eq_zero: ∀ (i : ℕ), ↑v i * a i ^ 2 = 0
i: ℕ
ha: a i ^ 2 = 0
inr
↑v i * 0 = 0]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq_summand: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), ↑v i * a i ^ 2 = 0
summand_nonneg: ∀ (i : ℕ), 0 ≤ ↑v i * a i ^ 2
summand_summable: Summable fun i => ↑v i * a i ^ 2
summand_eq_zero: ∀ (i : ℕ), ↑v i * a i ^ 2 = 0
i: ℕ
ha: a i ^ 2 = 0
inr
↑v i * 0 = 0
      
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq_summand: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), ↑v i * a i ^ 2 = 0
summand_nonneg: ∀ (i : ℕ), 0 ≤ ↑v i * a i ^ 2
summand_summable: Summable fun i => ↑v i * a i ^ 2
summand_eq_zero: ∀ (i : ℕ), ↑v i * a i ^ 2 = 0
i: ℕ
ha: a i ^ 2 = 0
inr
↑v i * a i = 0ring
Goals accomplished! 🐙
  }
Goals accomplished! 🐙

  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
⟪f, f⟫_ℝ = 0 → f = 0ext x
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq_summand: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), ↑v i * a i ^ 2 = 0
summand_nonneg: ∀ (i : ℕ), 0 ≤ ↑v i * a i ^ 2
summand_summable: Summable fun i => ↑v i * a i ^ 2
summand_eq_zero: ∀ (i : ℕ), ↑v i * a i ^ 2 = 0
mul_v_a_eq_0: ∀ (i : ℕ), ↑v i * a i = 0
x: ↑Ω
a.h
↑f x = ↑0 x
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
⟪f, f⟫_ℝ = 0 → f = 0rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq_summand: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), ↑v i * a i ^ 2 = 0
summand_nonneg: ∀ (i : ℕ), 0 ≤ ↑v i * a i ^ 2
summand_summable: Summable fun i => ↑v i * a i ^ 2
summand_eq_zero: ∀ (i : ℕ), ↑v i * a i ^ 2 = 0
mul_v_a_eq_0: ∀ (i : ℕ), ↑v i * a i = 0
x: ↑Ω
a.h
↑f x = ↑0 xshow (0 : H v e μ).1 = zero by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq_summand: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), ↑v i * a i ^ 2 = 0
summand_nonneg: ∀ (i : ℕ), 0 ≤ ↑v i * a i ^ 2
summand_summable: Summable fun i => ↑v i * a i ^ 2
summand_eq_zero: ∀ (i : ℕ), ↑v i * a i ^ 2 = 0
mul_v_a_eq_0: ∀ (i : ℕ), ↑v i * a i = 0
x: ↑Ω
a.h
↑f x = ↑0 xrfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq_summand: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), ↑v i * a i ^ 2 = 0
summand_nonneg: ∀ (i : ℕ), 0 ≤ ↑v i * a i ^ 2
summand_summable: Summable fun i => ↑v i * a i ^ 2
summand_eq_zero: ∀ (i : ℕ), ↑v i * a i ^ 2 = 0
mul_v_a_eq_0: ∀ (i : ℕ), ↑v i * a i = 0
x: ↑Ω
a.h
↑f x = zero x
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
⟪f, f⟫_ℝ = 0 → f = 0rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq_summand: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), ↑v i * a i ^ 2 = 0
summand_nonneg: ∀ (i : ℕ), 0 ≤ ↑v i * a i ^ 2
summand_summable: Summable fun i => ↑v i * a i ^ 2
summand_eq_zero: ∀ (i : ℕ), ↑v i * a i ^ 2 = 0
mul_v_a_eq_0: ∀ (i : ℕ), ↑v i * a i = 0
x: ↑Ω
a.h
↑f x = zero xshow zero x = 0 by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq_summand: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), ↑v i * a i ^ 2 = 0
summand_nonneg: ∀ (i : ℕ), 0 ≤ ↑v i * a i ^ 2
summand_summable: Summable fun i => ↑v i * a i ^ 2
summand_eq_zero: ∀ (i : ℕ), ↑v i * a i ^ 2 = 0
mul_v_a_eq_0: ∀ (i : ℕ), ↑v i * a i = 0
x: ↑Ω
a.h
↑f x = zero xrfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq_summand: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), ↑v i * a i ^ 2 = 0
summand_nonneg: ∀ (i : ℕ), 0 ≤ ↑v i * a i ^ 2
summand_summable: Summable fun i => ↑v i * a i ^ 2
summand_eq_zero: ∀ (i : ℕ), ↑v i * a i ^ 2 = 0
mul_v_a_eq_0: ∀ (i : ℕ), ↑v i * a i = 0
x: ↑Ω
a.h
↑f x = 0
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
⟪f, f⟫_ℝ = 0 → f = 0rcases (set_repr_ne f).some_mem with ⟨⟨ha_r, 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq_summand: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), ↑v i * a i ^ 2 = 0
summand_nonneg: ∀ (i : ℕ), 0 ≤ ↑v i * a i ^ 2
summand_summable: Summable fun i => ↑v i * a i ^ 2
summand_eq_zero: ∀ (i : ℕ), ↑v i * a i ^ 2 = 0
mul_v_a_eq_0: ∀ (i : ℕ), ↑v i * a i = 0
x: ↑Ω
a.h
↑f x = 0ha_s
Warning: unused variable `ha_s` [linter.unusedVariables]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq_summand: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), ↑v i * a i ^ 2 = 0
summand_nonneg: ∀ (i : ℕ), 0 ≤ ↑v i * a i ^ 2
summand_summable: Summable fun i => ↑v i * a i ^ 2
summand_eq_zero: ∀ (i : ℕ), ↑v i * a i ^ 2 = 0
mul_v_a_eq_0: ∀ (i : ℕ), ↑v i * a i = 0
x: ↑Ω
a.h
↑f x = 0⟩, _⟩
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq_summand: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), ↑v i * a i ^ 2 = 0
summand_nonneg: ∀ (i : ℕ), 0 ≤ ↑v i * a i ^ 2
summand_summable: Summable fun i => ↑v i * a i ^ 2
summand_eq_zero: ∀ (i : ℕ), ↑v i * a i ^ 2 = 0
mul_v_a_eq_0: ∀ (i : ℕ), ↑v i * a i = 0
x: ↑Ω
right✝: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
ha_r: ↑f = fun x => ∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * e i x
ha_s: ∀ (x : ↑Ω), Summable fun i => ↑v i * Set.Nonempty.some ⋯ i * e i x
a.h.intro.intro
↑f x = 0
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
⟪f, f⟫_ℝ = 0 → f = 0rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq_summand: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), ↑v i * a i ^ 2 = 0
summand_nonneg: ∀ (i : ℕ), 0 ≤ ↑v i * a i ^ 2
summand_summable: Summable fun i => ↑v i * a i ^ 2
summand_eq_zero: ∀ (i : ℕ), ↑v i * a i ^ 2 = 0
mul_v_a_eq_0: ∀ (i : ℕ), ↑v i * a i = 0
x: ↑Ω
right✝: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
ha_r: ↑f = fun x => ∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * e i x
ha_s: ∀ (x : ↑Ω), Summable fun i => ↑v i * Set.Nonempty.some ⋯ i * e i x
a.h.intro.intro
↑f x = 0ha_r,
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq_summand: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), ↑v i * a i ^ 2 = 0
summand_nonneg: ∀ (i : ℕ), 0 ≤ ↑v i * a i ^ 2
summand_summable: Summable fun i => ↑v i * a i ^ 2
summand_eq_zero: ∀ (i : ℕ), ↑v i * a i ^ 2 = 0
mul_v_a_eq_0: ∀ (i : ℕ), ↑v i * a i = 0
x: ↑Ω
right✝: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
ha_r: ↑f = fun x => ∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * e i x
ha_s: ∀ (x : ↑Ω), Summable fun i => ↑v i * Set.Nonempty.some ⋯ i * e i x
a.h.intro.intro
(fun x => ∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * e i x) x = 0 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq_summand: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), ↑v i * a i ^ 2 = 0
summand_nonneg: ∀ (i : ℕ), 0 ≤ ↑v i * a i ^ 2
summand_summable: Summable fun i => ↑v i * a i ^ 2
summand_eq_zero: ∀ (i : ℕ), ↑v i * a i ^ 2 = 0
mul_v_a_eq_0: ∀ (i : ℕ), ↑v i * a i = 0
x: ↑Ω
right✝: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
ha_r: ↑f = fun x => ∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * e i x
ha_s: ∀ (x : ↑Ω), Summable fun i => ↑v i * Set.Nonempty.some ⋯ i * e i x
a.h.intro.intro
↑f x = 0show (set_repr_ne f).some = a by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq_summand: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), ↑v i * a i ^ 2 = 0
summand_nonneg: ∀ (i : ℕ), 0 ≤ ↑v i * a i ^ 2
summand_summable: Summable fun i => ↑v i * a i ^ 2
summand_eq_zero: ∀ (i : ℕ), ↑v i * a i ^ 2 = 0
mul_v_a_eq_0: ∀ (i : ℕ), ↑v i * a i = 0
x: ↑Ω
right✝: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
ha_r: ↑f = fun x => ∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * e i x
ha_s: ∀ (x : ↑Ω), Summable fun i => ↑v i * Set.Nonempty.some ⋯ i * e i x
a.h.intro.intro
(fun x => ∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * e i x) x = 0rfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq_summand: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), ↑v i * a i ^ 2 = 0
summand_nonneg: ∀ (i : ℕ), 0 ≤ ↑v i * a i ^ 2
summand_summable: Summable fun i => ↑v i * a i ^ 2
summand_eq_zero: ∀ (i : ℕ), ↑v i * a i ^ 2 = 0
mul_v_a_eq_0: ∀ (i : ℕ), ↑v i * a i = 0
x: ↑Ω
right✝: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
ha_r: ↑f = fun x => ∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * e i x
ha_s: ∀ (x : ↑Ω), Summable fun i => ↑v i * Set.Nonempty.some ⋯ i * e i x
a.h.intro.intro
(fun x => ∑' (i : ℕ), ↑v i * a i * e i x) x = 0

  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
⟪f, f⟫_ℝ = 0 → f = 0have summand_eq_0 : ∀ i, (v.1 i) * (a i) * (e i x) = 0 := 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq_summand: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), ↑v i * a i ^ 2 = 0
summand_nonneg: ∀ (i : ℕ), 0 ≤ ↑v i * a i ^ 2
summand_summable: Summable fun i => ↑v i * a i ^ 2
summand_eq_zero: ∀ (i : ℕ), ↑v i * a i ^ 2 = 0
mul_v_a_eq_0: ∀ (i : ℕ), ↑v i * a i = 0
x: ↑Ω
right✝: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
ha_r: ↑f = fun x => ∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * e i x
ha_s: ∀ (x : ↑Ω), Summable fun i => ↑v i * Set.Nonempty.some ⋯ i * e i x
a.h.intro.intro
(fun x => ∑' (i : ℕ), ↑v i * a i * e i x) x = 0by
Goals accomplished! 🐙 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq_summand: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), ↑v i * a i ^ 2 = 0
summand_nonneg: ∀ (i : ℕ), 0 ≤ ↑v i * a i ^ 2
summand_summable: Summable fun i => ↑v i * a i ^ 2
summand_eq_zero: ∀ (i : ℕ), ↑v i * a i ^ 2 = 0
mul_v_a_eq_0: ∀ (i : ℕ), ↑v i * a i = 0
x: ↑Ω
right✝: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
ha_r: ↑f = fun x => ∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * e i x
ha_s: ∀ (x : ↑Ω), Summable fun i => ↑v i * Set.Nonempty.some ⋯ i * e i x
∀ (i : ℕ), ↑v i * a i * e i x = 0{
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq_summand: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), ↑v i * a i ^ 2 = 0
summand_nonneg: ∀ (i : ℕ), 0 ≤ ↑v i * a i ^ 2
summand_summable: Summable fun i => ↑v i * a i ^ 2
summand_eq_zero: ∀ (i : ℕ), ↑v i * a i ^ 2 = 0
mul_v_a_eq_0: ∀ (i : ℕ), ↑v i * a i = 0
x: ↑Ω
right✝: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
ha_r: ↑f = fun x => ∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * e i x
ha_s: ∀ (x : ↑Ω), Summable fun i => ↑v i * Set.Nonempty.some ⋯ i * e i x
∀ (i : ℕ), ↑v i * a i * e i x = 0
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq_summand: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), ↑v i * a i ^ 2 = 0
summand_nonneg: ∀ (i : ℕ), 0 ≤ ↑v i * a i ^ 2
summand_summable: Summable fun i => ↑v i * a i ^ 2
summand_eq_zero: ∀ (i : ℕ), ↑v i * a i ^ 2 = 0
mul_v_a_eq_0: ∀ (i : ℕ), ↑v i * a i = 0
x: ↑Ω
right✝: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
ha_r: ↑f = fun x => ∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * e i x
ha_s: ∀ (x : ↑Ω), Summable fun i => ↑v i * Set.Nonempty.some ⋯ i * e i x
∀ (i : ℕ), ↑v i * a i * e i x = 0intro i
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq_summand: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), ↑v i * a i ^ 2 = 0
summand_nonneg: ∀ (i : ℕ), 0 ≤ ↑v i * a i ^ 2
summand_summable: Summable fun i => ↑v i * a i ^ 2
summand_eq_zero: ∀ (i : ℕ), ↑v i * a i ^ 2 = 0
mul_v_a_eq_0: ∀ (i : ℕ), ↑v i * a i = 0
x: ↑Ω
right✝: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
ha_r: ↑f = fun x => ∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * e i x
ha_s: ∀ (x : ↑Ω), Summable fun i => ↑v i * Set.Nonempty.some ⋯ i * e i x
i: ℕ
↑v i * a i * e i x = 0
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq_summand: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), ↑v i * a i ^ 2 = 0
summand_nonneg: ∀ (i : ℕ), 0 ≤ ↑v i * a i ^ 2
summand_summable: Summable fun i => ↑v i * a i ^ 2
summand_eq_zero: ∀ (i : ℕ), ↑v i * a i ^ 2 = 0
mul_v_a_eq_0: ∀ (i : ℕ), ↑v i * a i = 0
x: ↑Ω
right✝: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
ha_r: ↑f = fun x => ∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * e i x
ha_s: ∀ (x : ↑Ω), Summable fun i => ↑v i * Set.Nonempty.some ⋯ i * e i x
∀ (i : ℕ), ↑v i * a i * e i x = 0rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq_summand: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), ↑v i * a i ^ 2 = 0
summand_nonneg: ∀ (i : ℕ), 0 ≤ ↑v i * a i ^ 2
summand_summable: Summable fun i => ↑v i * a i ^ 2
summand_eq_zero: ∀ (i : ℕ), ↑v i * a i ^ 2 = 0
mul_v_a_eq_0: ∀ (i : ℕ), ↑v i * a i = 0
x: ↑Ω
right✝: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
ha_r: ↑f = fun x => ∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * e i x
ha_s: ∀ (x : ↑Ω), Summable fun i => ↑v i * Set.Nonempty.some ⋯ i * e i x
i: ℕ
↑v i * a i * e i x = 0mul_v_a_eq_0
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq_summand: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), ↑v i * a i ^ 2 = 0
summand_nonneg: ∀ (i : ℕ), 0 ≤ ↑v i * a i ^ 2
summand_summable: Summable fun i => ↑v i * a i ^ 2
summand_eq_zero: ∀ (i : ℕ), ↑v i * a i ^ 2 = 0
mul_v_a_eq_0: ∀ (i : ℕ), ↑v i * a i = 0
x: ↑Ω
right✝: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
ha_r: ↑f = fun x => ∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * e i x
ha_s: ∀ (x : ↑Ω), Summable fun i => ↑v i * Set.Nonempty.some ⋯ i * e i x
i: ℕ
0 * e i x = 0]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq_summand: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), ↑v i * a i ^ 2 = 0
summand_nonneg: ∀ (i : ℕ), 0 ≤ ↑v i * a i ^ 2
summand_summable: Summable fun i => ↑v i * a i ^ 2
summand_eq_zero: ∀ (i : ℕ), ↑v i * a i ^ 2 = 0
mul_v_a_eq_0: ∀ (i : ℕ), ↑v i * a i = 0
x: ↑Ω
right✝: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
ha_r: ↑f = fun x => ∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * e i x
ha_s: ∀ (x : ↑Ω), Summable fun i => ↑v i * Set.Nonempty.some ⋯ i * e i x
i: ℕ
0 * e i x = 0
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq_summand: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), ↑v i * a i ^ 2 = 0
summand_nonneg: ∀ (i : ℕ), 0 ≤ ↑v i * a i ^ 2
summand_summable: Summable fun i => ↑v i * a i ^ 2
summand_eq_zero: ∀ (i : ℕ), ↑v i * a i ^ 2 = 0
mul_v_a_eq_0: ∀ (i : ℕ), ↑v i * a i = 0
x: ↑Ω
right✝: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
ha_r: ↑f = fun x => ∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * e i x
ha_s: ∀ (x : ↑Ω), Summable fun i => ↑v i * Set.Nonempty.some ⋯ i * e i x
∀ (i : ℕ), ↑v i * a i * e i x = 0ring
Goals accomplished! 🐙
  }
Goals accomplished! 🐙

  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
⟪f, f⟫_ℝ = 0 → f = 0simp_rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq_summand: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), ↑v i * a i ^ 2 = 0
summand_nonneg: ∀ (i : ℕ), 0 ≤ ↑v i * a i ^ 2
summand_summable: Summable fun i => ↑v i * a i ^ 2
summand_eq_zero: ∀ (i : ℕ), ↑v i * a i ^ 2 = 0
mul_v_a_eq_0: ∀ (i : ℕ), ↑v i * a i = 0
x: ↑Ω
right✝: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
ha_r: ↑f = fun x => ∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * e i x
ha_s: ∀ (x : ↑Ω), Summable fun i => ↑v i * Set.Nonempty.some ⋯ i * e i x
summand_eq_0: ∀ (i : ℕ), ↑v i * a i * e i x = 0
a.h.intro.intro
∑' (i : ℕ), ↑v i * a i * e i x = 0summand_eq_0
[Meta.Tactic.simp.rewrite] summand_eq_0:1000, ↑v i * a i * e i x ==> 0
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq_summand: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), ↑v i * a i ^ 2 = 0
summand_nonneg: ∀ (i : ℕ), 0 ≤ ↑v i * a i ^ 2
summand_summable: Summable fun i => ↑v i * a i ^ 2
summand_eq_zero: ∀ (i : ℕ), ↑v i * a i ^ 2 = 0
mul_v_a_eq_0: ∀ (i : ℕ), ↑v i * a i = 0
x: ↑Ω
right✝: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
ha_r: ↑f = fun x => ∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * e i x
ha_s: ∀ (x : ↑Ω), Summable fun i => ↑v i * Set.Nonempty.some ⋯ i * e i x
summand_eq_0: ∀ (i : ℕ), ↑v i * a i * e i x = 0
a.h.intro.intro
∑' (i : ℕ), 0 = 0]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
a:= Set.Nonempty.some ⋯: ℕ → ℝ
sq_summand: ∀ (i : ℕ), ↑v i * a i * a i = ↑v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), ↑v i * a i ^ 2 = 0
summand_nonneg: ∀ (i : ℕ), 0 ≤ ↑v i * a i ^ 2
summand_summable: Summable fun i => ↑v i * a i ^ 2
summand_eq_zero: ∀ (i : ℕ), ↑v i * a i ^ 2 = 0
mul_v_a_eq_0: ∀ (i : ℕ), ↑v i * a i = 0
x: ↑Ω
right✝: Summable fun i => ↑v i * Set.Nonempty.some ⋯ i ^ 2
ha_r: ↑f = fun x => ∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * e i x
ha_s: ∀ (x : ↑Ω), Summable fun i => ↑v i * Set.Nonempty.some ⋯ i * e i x
summand_eq_0: ∀ (i : ℕ), ↑v i * a i * e i x = 0
a.h.intro.intro
∑' (i : ℕ), 0 = 0
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
⟪f, f⟫_ℝ = 0 → f = 0exact tsum_zero
Goals accomplished! 🐙

lemma inner_eq_sq_norm : inner f f = (norm f)^2 := by
Goals accomplished! 🐙
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
⟪f, f⟫_ℝ = ‖f‖ ^ 2rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
⟪f, f⟫_ℝ = ‖f‖ ^ 2show norm f = Real.sqrt (inner f f) by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
⟪f, f⟫_ℝ = ‖f‖ ^ 2rfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
⟪f, f⟫_ℝ = √⟪f, f⟫_ℝ ^ 2
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
⟪f, f⟫_ℝ = ‖f‖ ^ 2rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
⟪f, f⟫_ℝ = √⟪f, f⟫_ℝ ^ 2Real.sq_sqrt (inner_nonneg f)
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
⟪f, f⟫_ℝ = ⟪f, f⟫_ℝ]
Goals accomplished! 🐙

lemma distrib_H_norm (t : ℝ) : ‖f + t*g‖^2 = ‖f‖^2 + (2 : ℝ) * t * inner f g + t^2 * ‖g‖^2 := by
Goals accomplished! 🐙
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
‖f + t * g‖ ^ 2 = ‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
‖f + t * g‖ ^ 2 = ‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2show  ‖f + t*g‖ = Real.sqrt (inner (f + t*g) (f + t*g)) by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
‖f + t * g‖ ^ 2 = ‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2rfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
√⟪f + t * g, f + t * g⟫_ℝ ^ 2 = ‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
‖f + t * g‖ ^ 2 = ‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2have inner_nn : (0 : ℝ) <= inner (f + t*g) (f + t*g) := inner_nonneg (f + t * g)
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
√⟪f + t * g, f + t * g⟫_ℝ ^ 2 = ‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
‖f + t * g‖ ^ 2 = ‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
√⟪f + t * g, f + t * g⟫_ℝ ^ 2 = ‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2Real.sq_sqrt inner_nn
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
⟪f + t * g, f + t * g⟫_ℝ = ‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
⟪f + t * g, f + t * g⟫_ℝ = ‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2

  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
‖f + t * g‖ ^ 2 = ‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2let a_f := (set_repr_ne f).some
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
⟪f + t * g, f + t * g⟫_ℝ = ‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
‖f + t * g‖ ^ 2 = ‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2let a_g := (set_repr_ne g).some
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
⟪f + t * g, f + t * g⟫_ℝ = ‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
‖f + t * g‖ ^ 2 = ‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2let a_f_tg := λ i ↦ a_f i + t * a_g i
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_f_tg:= fun i => a_f i + t * a_g i: ℕ → ℝ
⟪f + t * g, f + t * g⟫_ℝ = ‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2

  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
‖f + t * g‖ ^ 2 = ‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2have a_f_tg_repr : a_f_tg ∈ set_repr (f + t*g) := 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_f_tg:= fun i => a_f i + t * a_g i: ℕ → ℝ
⟪f + t * g, f + t * g⟫_ℝ = ‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2by
Goals accomplished! 🐙 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_f_tg:= fun i => a_f i + t * a_g i: ℕ → ℝ
a_f_tg ∈ set_repr (f + t * g){
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_f_tg:= fun i => a_f i + t * a_g i: ℕ → ℝ
a_f_tg ∈ set_repr (f + t * g)
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_f_tg:= fun i => a_f i + t * a_g i: ℕ → ℝ
a_f_tg ∈ set_repr (f + t * g)have repr_add :
      (λ i ↦ a_f i + (set_repr_ne (t*g)).some i) ∈ set_repr (f + t*g)
      := ⟨add_repr f (t*g), add_summable f (t*g)⟩
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_f_tg:= fun i => a_f i + t * a_g i: ℕ → ℝ
repr_add: (fun i => a_f i + Set.Nonempty.some ⋯ i) ∈ set_repr (f + t * g)
a_f_tg ∈ set_repr (f + t * g)
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_f_tg:= fun i => a_f i + t * a_g i: ℕ → ℝ
a_f_tg ∈ set_repr (f + t * g)have repr_mul :
      (λ i ↦ t*(a_g i)) ∈ set_repr (t*g)
      := ⟨mul_repr g t, mul_summable g t⟩
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_f_tg:= fun i => a_f i + t * a_g i: ℕ → ℝ
repr_add: (fun i => a_f i + Set.Nonempty.some ⋯ i) ∈ set_repr (f + t * g)
repr_mul: (fun i => t * a_g i) ∈ set_repr (t * g)
a_f_tg ∈ set_repr (f + t * g)

    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_f_tg:= fun i => a_f i + t * a_g i: ℕ → ℝ
a_f_tg ∈ set_repr (f + t * g)rwa [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_f_tg:= fun i => a_f i + t * a_g i: ℕ → ℝ
repr_add: (fun i => a_f i + Set.Nonempty.some ⋯ i) ∈ set_repr (f + t * g)
repr_mul: (fun i => t * a_g i) ∈ set_repr (t * g)
a_f_tg ∈ set_repr (f + t * g)unique_choice repr_mul
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_f_tg:= fun i => a_f i + t * a_g i: ℕ → ℝ
repr_add: (fun i => a_f i + (fun i => t * a_g i) i) ∈ set_repr (f + t * g)
repr_mul: (fun i => t * a_g i) ∈ set_repr (t * g)
a_f_tg ∈ set_repr (f + t * g)]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_f_tg:= fun i => a_f i + t * a_g i: ℕ → ℝ
repr_add: (fun i => a_f i + (fun i => t * a_g i) i) ∈ set_repr (f + t * g)
repr_mul: (fun i => t * a_g i) ∈ set_repr (t * g)
a_f_tg ∈ set_repr (f + t * g) at repr_add
Goals accomplished! 🐙
  }
Goals accomplished! 🐙

  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
‖f + t * g‖ ^ 2 = ‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_f_tg:= fun i => a_f i + t * a_g i: ℕ → ℝ
a_f_tg_repr: a_f_tg ∈ set_repr (f + t * g)
⟪f + t * g, f + t * g⟫_ℝ = ‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2show inner (f + t*g) (f + t*g) = H_inner (f + t*g) (f + t*g) by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_f_tg:= fun i => a_f i + t * a_g i: ℕ → ℝ
a_f_tg_repr: a_f_tg ∈ set_repr (f + t * g)
⟪f + t * g, f + t * g⟫_ℝ = ‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2rfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_f_tg:= fun i => a_f i + t * a_g i: ℕ → ℝ
a_f_tg_repr: a_f_tg ∈ set_repr (f + t * g)
H_inner (f + t * g) (f + t * g) = ‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
‖f + t * g‖ ^ 2 = ‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2unfold H_inner
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_f_tg:= fun i => a_f i + t * a_g i: ℕ → ℝ
a_f_tg_repr: a_f_tg ∈ set_repr (f + t * g)
∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * Set.Nonempty.some ⋯ i = ‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
‖f + t * g‖ ^ 2 = ‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_f_tg:= fun i => a_f i + t * a_g i: ℕ → ℝ
a_f_tg_repr: a_f_tg ∈ set_repr (f + t * g)
∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * Set.Nonempty.some ⋯ i = ‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2unique_choice a_f_tg_repr
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_f_tg:= fun i => a_f i + t * a_g i: ℕ → ℝ
a_f_tg_repr: a_f_tg ∈ set_repr (f + t * g)
∑' (i : ℕ), ↑v i * a_f_tg i * a_f_tg i = ‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_f_tg:= fun i => a_f i + t * a_g i: ℕ → ℝ
a_f_tg_repr: a_f_tg ∈ set_repr (f + t * g)
∑' (i : ℕ), ↑v i * a_f_tg i * a_f_tg i = ‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2

  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
‖f + t * g‖ ^ 2 = ‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2have distribute : ∀ i, v.1 i * a_f_tg i * a_f_tg i = v.1 i * a_f i * a_f i + (2:ℝ) * t * (v.1 i * a_f i * a_g i) + t^2 * (v.1 i * a_g i * a_g i) := 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_f_tg:= fun i => a_f i + t * a_g i: ℕ → ℝ
a_f_tg_repr: a_f_tg ∈ set_repr (f + t * g)
∑' (i : ℕ), ↑v i * a_f_tg i * a_f_tg i = ‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2by
Goals accomplished! 🐙 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_f_tg:= fun i => a_f i + t * a_g i: ℕ → ℝ
a_f_tg_repr: a_f_tg ∈ set_repr (f + t * g)
∀ (i : ℕ),
  ↑v i * a_f_tg i * a_f_tg i = ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i) + t ^ 2 * (↑v i * a_g i * a_g i)intro i
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_f_tg:= fun i => a_f i + t * a_g i: ℕ → ℝ
a_f_tg_repr: a_f_tg ∈ set_repr (f + t * g)
i: ℕ
↑v i * a_f_tg i * a_f_tg i = ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i) + t ^ 2 * (↑v i * a_g i * a_g i);
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_f_tg:= fun i => a_f i + t * a_g i: ℕ → ℝ
a_f_tg_repr: a_f_tg ∈ set_repr (f + t * g)
i: ℕ
↑v i * a_f_tg i * a_f_tg i = ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i) + t ^ 2 * (↑v i * a_g i * a_g i) 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_f_tg:= fun i => a_f i + t * a_g i: ℕ → ℝ
a_f_tg_repr: a_f_tg ∈ set_repr (f + t * g)
∀ (i : ℕ),
  ↑v i * a_f_tg i * a_f_tg i = ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i) + t ^ 2 * (↑v i * a_g i * a_g i)ring
Goals accomplished! 🐙
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
‖f + t * g‖ ^ 2 = ‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2simp_rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_f_tg:= fun i => a_f i + t * a_g i: ℕ → ℝ
a_f_tg_repr: a_f_tg ∈ set_repr (f + t * g)
distribute: ∀ (i : ℕ),
  ↑v i * a_f_tg i * a_f_tg i = ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i) + t ^ 2 * (↑v i * a_g i * a_g i)
∑' (i : ℕ), ↑v i * a_f_tg i * a_f_tg i = ‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2distribute
[Meta.Tactic.simp.rewrite] distribute:1000, ↑v i * a_f_tg i *
      a_f_tg i ==> ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i) + t ^ 2 * (↑v i * a_g i * a_g i)
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_f_tg:= fun i => a_f i + t * a_g i: ℕ → ℝ
a_f_tg_repr: a_f_tg ∈ set_repr (f + t * g)
distribute: ∀ (i : ℕ),
  ↑v i * a_f_tg i * a_f_tg i = ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i) + t ^ 2 * (↑v i * a_g i * a_g i)
∑' (i : ℕ), (↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i) + t ^ 2 * (↑v i * a_g i * a_g i)) =
  ‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_f_tg:= fun i => a_f i + t * a_g i: ℕ → ℝ
a_f_tg_repr: a_f_tg ∈ set_repr (f + t * g)
distribute: ∀ (i : ℕ),
  ↑v i * a_f_tg i * a_f_tg i = ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i) + t ^ 2 * (↑v i * a_g i * a_g i)
∑' (i : ℕ), (↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i) + t ^ 2 * (↑v i * a_g i * a_g i)) =
  ‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2

  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
‖f + t * g‖ ^ 2 = ‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2have add_summable : Summable (λ i ↦ v.1 i * a_f i * a_f i + ((2:ℝ) * t) * (v.1 i * a_f i * a_g i)) :=(product_summable f f).add ((product_summable f g).mul_left ((2:ℝ) * t))
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_f_tg:= fun i => a_f i + t * a_g i: ℕ → ℝ
a_f_tg_repr: a_f_tg ∈ set_repr (f + t * g)
distribute: ∀ (i : ℕ),
  ↑v i * a_f_tg i * a_f_tg i = ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i) + t ^ 2 * (↑v i * a_g i * a_g i)
add_summable: Summable fun i => ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i)
∑' (i : ℕ), (↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i) + t ^ 2 * (↑v i * a_g i * a_g i)) =
  ‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2

  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
‖f + t * g‖ ^ 2 = ‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_f_tg:= fun i => a_f i + t * a_g i: ℕ → ℝ
a_f_tg_repr: a_f_tg ∈ set_repr (f + t * g)
distribute: ∀ (i : ℕ),
  ↑v i * a_f_tg i * a_f_tg i = ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i) + t ^ 2 * (↑v i * a_g i * a_g i)
add_summable: Summable fun i => ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i)
∑' (i : ℕ), (↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i) + t ^ 2 * (↑v i * a_g i * a_g i)) =
  ‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2tsum_add add_summable ((product_summable g g).mul_left (t^2))
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_f_tg:= fun i => a_f i + t * a_g i: ℕ → ℝ
a_f_tg_repr: a_f_tg ∈ set_repr (f + t * g)
distribute: ∀ (i : ℕ),
  ↑v i * a_f_tg i * a_f_tg i = ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i) + t ^ 2 * (↑v i * a_g i * a_g i)
add_summable: Summable fun i => ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i)
∑' (b : ℕ), (↑v b * a_f b * a_f b + 2 * t * (↑v b * a_f b * a_g b)) +
    ∑' (b : ℕ), t ^ 2 * (↑v b * Set.Nonempty.some ⋯ b * Set.Nonempty.some ⋯ b) =
  ‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_f_tg:= fun i => a_f i + t * a_g i: ℕ → ℝ
a_f_tg_repr: a_f_tg ∈ set_repr (f + t * g)
distribute: ∀ (i : ℕ),
  ↑v i * a_f_tg i * a_f_tg i = ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i) + t ^ 2 * (↑v i * a_g i * a_g i)
add_summable: Summable fun i => ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i)
∑' (b : ℕ), (↑v b * a_f b * a_f b + 2 * t * (↑v b * a_f b * a_g b)) +
    ∑' (b : ℕ), t ^ 2 * (↑v b * Set.Nonempty.some ⋯ b * Set.Nonempty.some ⋯ b) =
  ‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2

  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
‖f + t * g‖ ^ 2 = ‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2have tsum_to_norm (h : H v e μ) : ∑' i, v.1 i * (set_repr_ne h).some i * (set_repr_ne h).some i = (norm h)^2 := 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_f_tg:= fun i => a_f i + t * a_g i: ℕ → ℝ
a_f_tg_repr: a_f_tg ∈ set_repr (f + t * g)
distribute: ∀ (i : ℕ),
  ↑v i * a_f_tg i * a_f_tg i = ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i) + t ^ 2 * (↑v i * a_g i * a_g i)
add_summable: Summable fun i => ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i)
∑' (b : ℕ), (↑v b * a_f b * a_f b + 2 * t * (↑v b * a_f b * a_g b)) +
    ∑' (b : ℕ), t ^ 2 * (↑v b * Set.Nonempty.some ⋯ b * Set.Nonempty.some ⋯ b) =
  ‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2by
Goals accomplished! 🐙 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_f_tg:= fun i => a_f i + t * a_g i: ℕ → ℝ
a_f_tg_repr: a_f_tg ∈ set_repr (f + t * g)
distribute: ∀ (i : ℕ),
  ↑v i * a_f_tg i * a_f_tg i = ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i) + t ^ 2 * (↑v i * a_g i * a_g i)
add_summable: Summable fun i => ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i)
h: ↑(H v e μ)
∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * Set.Nonempty.some ⋯ i = ‖h‖ ^ 2{
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_f_tg:= fun i => a_f i + t * a_g i: ℕ → ℝ
a_f_tg_repr: a_f_tg ∈ set_repr (f + t * g)
distribute: ∀ (i : ℕ),
  ↑v i * a_f_tg i * a_f_tg i = ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i) + t ^ 2 * (↑v i * a_g i * a_g i)
add_summable: Summable fun i => ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i)
h: ↑(H v e μ)
∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * Set.Nonempty.some ⋯ i = ‖h‖ ^ 2
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_f_tg:= fun i => a_f i + t * a_g i: ℕ → ℝ
a_f_tg_repr: a_f_tg ∈ set_repr (f + t * g)
distribute: ∀ (i : ℕ),
  ↑v i * a_f_tg i * a_f_tg i = ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i) + t ^ 2 * (↑v i * a_g i * a_g i)
add_summable: Summable fun i => ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i)
h: ↑(H v e μ)
∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * Set.Nonempty.some ⋯ i = ‖h‖ ^ 2rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_f_tg:= fun i => a_f i + t * a_g i: ℕ → ℝ
a_f_tg_repr: a_f_tg ∈ set_repr (f + t * g)
distribute: ∀ (i : ℕ),
  ↑v i * a_f_tg i * a_f_tg i = ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i) + t ^ 2 * (↑v i * a_g i * a_g i)
add_summable: Summable fun i => ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i)
h: ↑(H v e μ)
∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * Set.Nonempty.some ⋯ i = ‖h‖ ^ 2show ∑' i, v.1 i * (set_repr_ne h).some i * (set_repr_ne h).some i = H_inner h h by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_f_tg:= fun i => a_f i + t * a_g i: ℕ → ℝ
a_f_tg_repr: a_f_tg ∈ set_repr (f + t * g)
distribute: ∀ (i : ℕ),
  ↑v i * a_f_tg i * a_f_tg i = ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i) + t ^ 2 * (↑v i * a_g i * a_g i)
add_summable: Summable fun i => ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i)
h: ↑(H v e μ)
∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * Set.Nonempty.some ⋯ i = ‖h‖ ^ 2rfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_f_tg:= fun i => a_f i + t * a_g i: ℕ → ℝ
a_f_tg_repr: a_f_tg ∈ set_repr (f + t * g)
distribute: ∀ (i : ℕ),
  ↑v i * a_f_tg i * a_f_tg i = ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i) + t ^ 2 * (↑v i * a_g i * a_g i)
add_summable: Summable fun i => ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i)
h: ↑(H v e μ)
H_inner h h = ‖h‖ ^ 2
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_f_tg:= fun i => a_f i + t * a_g i: ℕ → ℝ
a_f_tg_repr: a_f_tg ∈ set_repr (f + t * g)
distribute: ∀ (i : ℕ),
  ↑v i * a_f_tg i * a_f_tg i = ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i) + t ^ 2 * (↑v i * a_g i * a_g i)
add_summable: Summable fun i => ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i)
h: ↑(H v e μ)
∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * Set.Nonempty.some ⋯ i = ‖h‖ ^ 2rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_f_tg:= fun i => a_f i + t * a_g i: ℕ → ℝ
a_f_tg_repr: a_f_tg ∈ set_repr (f + t * g)
distribute: ∀ (i : ℕ),
  ↑v i * a_f_tg i * a_f_tg i = ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i) + t ^ 2 * (↑v i * a_g i * a_g i)
add_summable: Summable fun i => ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i)
h: ↑(H v e μ)
H_inner h h = ‖h‖ ^ 2show H_inner h h = inner h h by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_f_tg:= fun i => a_f i + t * a_g i: ℕ → ℝ
a_f_tg_repr: a_f_tg ∈ set_repr (f + t * g)
distribute: ∀ (i : ℕ),
  ↑v i * a_f_tg i * a_f_tg i = ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i) + t ^ 2 * (↑v i * a_g i * a_g i)
add_summable: Summable fun i => ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i)
h: ↑(H v e μ)
H_inner h h = ‖h‖ ^ 2rfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_f_tg:= fun i => a_f i + t * a_g i: ℕ → ℝ
a_f_tg_repr: a_f_tg ∈ set_repr (f + t * g)
distribute: ∀ (i : ℕ),
  ↑v i * a_f_tg i * a_f_tg i = ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i) + t ^ 2 * (↑v i * a_g i * a_g i)
add_summable: Summable fun i => ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i)
h: ↑(H v e μ)
⟪h, h⟫_ℝ = ‖h‖ ^ 2
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_f_tg:= fun i => a_f i + t * a_g i: ℕ → ℝ
a_f_tg_repr: a_f_tg ∈ set_repr (f + t * g)
distribute: ∀ (i : ℕ),
  ↑v i * a_f_tg i * a_f_tg i = ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i) + t ^ 2 * (↑v i * a_g i * a_g i)
add_summable: Summable fun i => ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i)
h: ↑(H v e μ)
∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * Set.Nonempty.some ⋯ i = ‖h‖ ^ 2rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_f_tg:= fun i => a_f i + t * a_g i: ℕ → ℝ
a_f_tg_repr: a_f_tg ∈ set_repr (f + t * g)
distribute: ∀ (i : ℕ),
  ↑v i * a_f_tg i * a_f_tg i = ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i) + t ^ 2 * (↑v i * a_g i * a_g i)
add_summable: Summable fun i => ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i)
h: ↑(H v e μ)
⟪h, h⟫_ℝ = ‖h‖ ^ 2inner_eq_sq_norm h
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_f_tg:= fun i => a_f i + t * a_g i: ℕ → ℝ
a_f_tg_repr: a_f_tg ∈ set_repr (f + t * g)
distribute: ∀ (i : ℕ),
  ↑v i * a_f_tg i * a_f_tg i = ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i) + t ^ 2 * (↑v i * a_g i * a_g i)
add_summable: Summable fun i => ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i)
h: ↑(H v e μ)
‖h‖ ^ 2 = ‖h‖ ^ 2]
Goals accomplished! 🐙
  }
Goals accomplished! 🐙

  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
‖f + t * g‖ ^ 2 = ‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_f_tg:= fun i => a_f i + t * a_g i: ℕ → ℝ
a_f_tg_repr: a_f_tg ∈ set_repr (f + t * g)
distribute: ∀ (i : ℕ),
  ↑v i * a_f_tg i * a_f_tg i = ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i) + t ^ 2 * (↑v i * a_g i * a_g i)
add_summable: Summable fun i => ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i)
tsum_to_norm: ∀ (h : ↑(H v e μ)), ∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * Set.Nonempty.some ⋯ i = ‖h‖ ^ 2
∑' (b : ℕ), (↑v b * a_f b * a_f b + 2 * t * (↑v b * a_f b * a_g b)) +
    ∑' (b : ℕ), t ^ 2 * (↑v b * Set.Nonempty.some ⋯ b * Set.Nonempty.some ⋯ b) =
  ‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2tsum_add (product_summable f f) ((product_summable f g).mul_left ((2:ℝ) * t))
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_f_tg:= fun i => a_f i + t * a_g i: ℕ → ℝ
a_f_tg_repr: a_f_tg ∈ set_repr (f + t * g)
distribute: ∀ (i : ℕ),
  ↑v i * a_f_tg i * a_f_tg i = ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i) + t ^ 2 * (↑v i * a_g i * a_g i)
add_summable: Summable fun i => ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i)
tsum_to_norm: ∀ (h : ↑(H v e μ)), ∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * Set.Nonempty.some ⋯ i = ‖h‖ ^ 2
∑' (b : ℕ), ↑v b * Set.Nonempty.some ⋯ b * Set.Nonempty.some ⋯ b +
      ∑' (b : ℕ), 2 * t * (↑v b * Set.Nonempty.some ⋯ b * Set.Nonempty.some ⋯ b) +
    ∑' (b : ℕ), t ^ 2 * (↑v b * Set.Nonempty.some ⋯ b * Set.Nonempty.some ⋯ b) =
  ‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_f_tg:= fun i => a_f i + t * a_g i: ℕ → ℝ
a_f_tg_repr: a_f_tg ∈ set_repr (f + t * g)
distribute: ∀ (i : ℕ),
  ↑v i * a_f_tg i * a_f_tg i = ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i) + t ^ 2 * (↑v i * a_g i * a_g i)
add_summable: Summable fun i => ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i)
tsum_to_norm: ∀ (h : ↑(H v e μ)), ∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * Set.Nonempty.some ⋯ i = ‖h‖ ^ 2
∑' (b : ℕ), ↑v b * Set.Nonempty.some ⋯ b * Set.Nonempty.some ⋯ b +
      ∑' (b : ℕ), 2 * t * (↑v b * Set.Nonempty.some ⋯ b * Set.Nonempty.some ⋯ b) +
    ∑' (b : ℕ), t ^ 2 * (↑v b * Set.Nonempty.some ⋯ b * Set.Nonempty.some ⋯ b) =
  ‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2

  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
‖f + t * g‖ ^ 2 = ‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2have const_out : ∑' i, t^2 * (v.1 i * a_g i * a_g i) =  t^2 * ∑' i, v.1 i * a_g i * a_g i := tsum_mul_left
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_f_tg:= fun i => a_f i + t * a_g i: ℕ → ℝ
a_f_tg_repr: a_f_tg ∈ set_repr (f + t * g)
distribute: ∀ (i : ℕ),
  ↑v i * a_f_tg i * a_f_tg i = ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i) + t ^ 2 * (↑v i * a_g i * a_g i)
add_summable: Summable fun i => ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i)
tsum_to_norm: ∀ (h : ↑(H v e μ)), ∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * Set.Nonempty.some ⋯ i = ‖h‖ ^ 2
const_out: ∑' (i : ℕ), t ^ 2 * (↑v i * a_g i * a_g i) = t ^ 2 * ∑' (i : ℕ), ↑v i * a_g i * a_g i
∑' (b : ℕ), ↑v b * Set.Nonempty.some ⋯ b * Set.Nonempty.some ⋯ b +
      ∑' (b : ℕ), 2 * t * (↑v b * Set.Nonempty.some ⋯ b * Set.Nonempty.some ⋯ b) +
    ∑' (b : ℕ), t ^ 2 * (↑v b * Set.Nonempty.some ⋯ b * Set.Nonempty.some ⋯ b) =
  ‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
‖f + t * g‖ ^ 2 = ‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_f_tg:= fun i => a_f i + t * a_g i: ℕ → ℝ
a_f_tg_repr: a_f_tg ∈ set_repr (f + t * g)
distribute: ∀ (i : ℕ),
  ↑v i * a_f_tg i * a_f_tg i = ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i) + t ^ 2 * (↑v i * a_g i * a_g i)
add_summable: Summable fun i => ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i)
tsum_to_norm: ∀ (h : ↑(H v e μ)), ∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * Set.Nonempty.some ⋯ i = ‖h‖ ^ 2
const_out: ∑' (i : ℕ), t ^ 2 * (↑v i * a_g i * a_g i) = t ^ 2 * ∑' (i : ℕ), ↑v i * a_g i * a_g i
∑' (b : ℕ), ↑v b * Set.Nonempty.some ⋯ b * Set.Nonempty.some ⋯ b +
      ∑' (b : ℕ), 2 * t * (↑v b * Set.Nonempty.some ⋯ b * Set.Nonempty.some ⋯ b) +
    ∑' (b : ℕ), t ^ 2 * (↑v b * Set.Nonempty.some ⋯ b * Set.Nonempty.some ⋯ b) =
  ‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2const_out,
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_f_tg:= fun i => a_f i + t * a_g i: ℕ → ℝ
a_f_tg_repr: a_f_tg ∈ set_repr (f + t * g)
distribute: ∀ (i : ℕ),
  ↑v i * a_f_tg i * a_f_tg i = ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i) + t ^ 2 * (↑v i * a_g i * a_g i)
add_summable: Summable fun i => ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i)
tsum_to_norm: ∀ (h : ↑(H v e μ)), ∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * Set.Nonempty.some ⋯ i = ‖h‖ ^ 2
const_out: ∑' (i : ℕ), t ^ 2 * (↑v i * a_g i * a_g i) = t ^ 2 * ∑' (i : ℕ), ↑v i * a_g i * a_g i
∑' (b : ℕ), ↑v b * Set.Nonempty.some ⋯ b * Set.Nonempty.some ⋯ b +
      ∑' (b : ℕ), 2 * t * (↑v b * Set.Nonempty.some ⋯ b * Set.Nonempty.some ⋯ b) +
    t ^ 2 * ∑' (i : ℕ), ↑v i * a_g i * a_g i =
  ‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_f_tg:= fun i => a_f i + t * a_g i: ℕ → ℝ
a_f_tg_repr: a_f_tg ∈ set_repr (f + t * g)
distribute: ∀ (i : ℕ),
  ↑v i * a_f_tg i * a_f_tg i = ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i) + t ^ 2 * (↑v i * a_g i * a_g i)
add_summable: Summable fun i => ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i)
tsum_to_norm: ∀ (h : ↑(H v e μ)), ∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * Set.Nonempty.some ⋯ i = ‖h‖ ^ 2
const_out: ∑' (i : ℕ), t ^ 2 * (↑v i * a_g i * a_g i) = t ^ 2 * ∑' (i : ℕ), ↑v i * a_g i * a_g i
∑' (b : ℕ), ↑v b * Set.Nonempty.some ⋯ b * Set.Nonempty.some ⋯ b +
      ∑' (b : ℕ), 2 * t * (↑v b * Set.Nonempty.some ⋯ b * Set.Nonempty.some ⋯ b) +
    ∑' (b : ℕ), t ^ 2 * (↑v b * Set.Nonempty.some ⋯ b * Set.Nonempty.some ⋯ b) =
  ‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2tsum_to_norm f,
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_f_tg:= fun i => a_f i + t * a_g i: ℕ → ℝ
a_f_tg_repr: a_f_tg ∈ set_repr (f + t * g)
distribute: ∀ (i : ℕ),
  ↑v i * a_f_tg i * a_f_tg i = ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i) + t ^ 2 * (↑v i * a_g i * a_g i)
add_summable: Summable fun i => ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i)
tsum_to_norm: ∀ (h : ↑(H v e μ)), ∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * Set.Nonempty.some ⋯ i = ‖h‖ ^ 2
const_out: ∑' (i : ℕ), t ^ 2 * (↑v i * a_g i * a_g i) = t ^ 2 * ∑' (i : ℕ), ↑v i * a_g i * a_g i
‖f‖ ^ 2 + ∑' (b : ℕ), 2 * t * (↑v b * Set.Nonempty.some ⋯ b * Set.Nonempty.some ⋯ b) +
    t ^ 2 * ∑' (i : ℕ), ↑v i * a_g i * a_g i =
  ‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_f_tg:= fun i => a_f i + t * a_g i: ℕ → ℝ
a_f_tg_repr: a_f_tg ∈ set_repr (f + t * g)
distribute: ∀ (i : ℕ),
  ↑v i * a_f_tg i * a_f_tg i = ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i) + t ^ 2 * (↑v i * a_g i * a_g i)
add_summable: Summable fun i => ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i)
tsum_to_norm: ∀ (h : ↑(H v e μ)), ∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * Set.Nonempty.some ⋯ i = ‖h‖ ^ 2
const_out: ∑' (i : ℕ), t ^ 2 * (↑v i * a_g i * a_g i) = t ^ 2 * ∑' (i : ℕ), ↑v i * a_g i * a_g i
∑' (b : ℕ), ↑v b * Set.Nonempty.some ⋯ b * Set.Nonempty.some ⋯ b +
      ∑' (b : ℕ), 2 * t * (↑v b * Set.Nonempty.some ⋯ b * Set.Nonempty.some ⋯ b) +
    ∑' (b : ℕ), t ^ 2 * (↑v b * Set.Nonempty.some ⋯ b * Set.Nonempty.some ⋯ b) =
  ‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2tsum_to_norm g
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_f_tg:= fun i => a_f i + t * a_g i: ℕ → ℝ
a_f_tg_repr: a_f_tg ∈ set_repr (f + t * g)
distribute: ∀ (i : ℕ),
  ↑v i * a_f_tg i * a_f_tg i = ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i) + t ^ 2 * (↑v i * a_g i * a_g i)
add_summable: Summable fun i => ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i)
tsum_to_norm: ∀ (h : ↑(H v e μ)), ∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * Set.Nonempty.some ⋯ i = ‖h‖ ^ 2
const_out: ∑' (i : ℕ), t ^ 2 * (↑v i * a_g i * a_g i) = t ^ 2 * ∑' (i : ℕ), ↑v i * a_g i * a_g i
‖f‖ ^ 2 + ∑' (b : ℕ), 2 * t * (↑v b * Set.Nonempty.some ⋯ b * Set.Nonempty.some ⋯ b) + t ^ 2 * ‖g‖ ^ 2 =
  ‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_f_tg:= fun i => a_f i + t * a_g i: ℕ → ℝ
a_f_tg_repr: a_f_tg ∈ set_repr (f + t * g)
distribute: ∀ (i : ℕ),
  ↑v i * a_f_tg i * a_f_tg i = ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i) + t ^ 2 * (↑v i * a_g i * a_g i)
add_summable: Summable fun i => ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i)
tsum_to_norm: ∀ (h : ↑(H v e μ)), ∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * Set.Nonempty.some ⋯ i = ‖h‖ ^ 2
const_out: ∑' (i : ℕ), t ^ 2 * (↑v i * a_g i * a_g i) = t ^ 2 * ∑' (i : ℕ), ↑v i * a_g i * a_g i
‖f‖ ^ 2 + ∑' (b : ℕ), 2 * t * (↑v b * Set.Nonempty.some ⋯ b * Set.Nonempty.some ⋯ b) + t ^ 2 * ‖g‖ ^ 2 =
  ‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2

  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
‖f + t * g‖ ^ 2 = ‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2have const_out : ∑' i, 2 * t * (v.1 i * a_f i * a_g i) =  2 * t * ∑' i, (v.1 i * a_f i * a_g i) := tsum_mul_left
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_f_tg:= fun i => a_f i + t * a_g i: ℕ → ℝ
a_f_tg_repr: a_f_tg ∈ set_repr (f + t * g)
distribute: ∀ (i : ℕ),
  ↑v i * a_f_tg i * a_f_tg i = ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i) + t ^ 2 * (↑v i * a_g i * a_g i)
add_summable: Summable fun i => ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i)
tsum_to_norm: ∀ (h : ↑(H v e μ)), ∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * Set.Nonempty.some ⋯ i = ‖h‖ ^ 2
const_out✝: ∑' (i : ℕ), t ^ 2 * (↑v i * a_g i * a_g i) = t ^ 2 * ∑' (i : ℕ), ↑v i * a_g i * a_g i
const_out: ∑' (i : ℕ), 2 * t * (↑v i * a_f i * a_g i) = 2 * t * ∑' (i : ℕ), ↑v i * a_f i * a_g i
‖f‖ ^ 2 + ∑' (b : ℕ), 2 * t * (↑v b * Set.Nonempty.some ⋯ b * Set.Nonempty.some ⋯ b) + t ^ 2 * ‖g‖ ^ 2 =
  ‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
‖f + t * g‖ ^ 2 = ‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_f_tg:= fun i => a_f i + t * a_g i: ℕ → ℝ
a_f_tg_repr: a_f_tg ∈ set_repr (f + t * g)
distribute: ∀ (i : ℕ),
  ↑v i * a_f_tg i * a_f_tg i = ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i) + t ^ 2 * (↑v i * a_g i * a_g i)
add_summable: Summable fun i => ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i)
tsum_to_norm: ∀ (h : ↑(H v e μ)), ∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * Set.Nonempty.some ⋯ i = ‖h‖ ^ 2
const_out✝: ∑' (i : ℕ), t ^ 2 * (↑v i * a_g i * a_g i) = t ^ 2 * ∑' (i : ℕ), ↑v i * a_g i * a_g i
const_out: ∑' (i : ℕ), 2 * t * (↑v i * a_f i * a_g i) = 2 * t * ∑' (i : ℕ), ↑v i * a_f i * a_g i
‖f‖ ^ 2 + ∑' (b : ℕ), 2 * t * (↑v b * Set.Nonempty.some ⋯ b * Set.Nonempty.some ⋯ b) + t ^ 2 * ‖g‖ ^ 2 =
  ‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2const_out
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_f_tg:= fun i => a_f i + t * a_g i: ℕ → ℝ
a_f_tg_repr: a_f_tg ∈ set_repr (f + t * g)
distribute: ∀ (i : ℕ),
  ↑v i * a_f_tg i * a_f_tg i = ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i) + t ^ 2 * (↑v i * a_g i * a_g i)
add_summable: Summable fun i => ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i)
tsum_to_norm: ∀ (h : ↑(H v e μ)), ∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * Set.Nonempty.some ⋯ i = ‖h‖ ^ 2
const_out✝: ∑' (i : ℕ), t ^ 2 * (↑v i * a_g i * a_g i) = t ^ 2 * ∑' (i : ℕ), ↑v i * a_g i * a_g i
const_out: ∑' (i : ℕ), 2 * t * (↑v i * a_f i * a_g i) = 2 * t * ∑' (i : ℕ), ↑v i * a_f i * a_g i
‖f‖ ^ 2 + 2 * t * ∑' (i : ℕ), ↑v i * a_f i * a_g i + t ^ 2 * ‖g‖ ^ 2 = ‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_f_tg:= fun i => a_f i + t * a_g i: ℕ → ℝ
a_f_tg_repr: a_f_tg ∈ set_repr (f + t * g)
distribute: ∀ (i : ℕ),
  ↑v i * a_f_tg i * a_f_tg i = ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i) + t ^ 2 * (↑v i * a_g i * a_g i)
add_summable: Summable fun i => ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i)
tsum_to_norm: ∀ (h : ↑(H v e μ)), ∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * Set.Nonempty.some ⋯ i = ‖h‖ ^ 2
const_out✝: ∑' (i : ℕ), t ^ 2 * (↑v i * a_g i * a_g i) = t ^ 2 * ∑' (i : ℕ), ↑v i * a_g i * a_g i
const_out: ∑' (i : ℕ), 2 * t * (↑v i * a_f i * a_g i) = 2 * t * ∑' (i : ℕ), ↑v i * a_f i * a_g i
‖f‖ ^ 2 + 2 * t * ∑' (i : ℕ), ↑v i * a_f i * a_g i + t ^ 2 * ‖g‖ ^ 2 = ‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2

  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
‖f + t * g‖ ^ 2 = ‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2have tsum_to_inner : ∑' i, v.1 i * a_f i * a_g i = inner f g := 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_f_tg:= fun i => a_f i + t * a_g i: ℕ → ℝ
a_f_tg_repr: a_f_tg ∈ set_repr (f + t * g)
distribute: ∀ (i : ℕ),
  ↑v i * a_f_tg i * a_f_tg i = ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i) + t ^ 2 * (↑v i * a_g i * a_g i)
add_summable: Summable fun i => ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i)
tsum_to_norm: ∀ (h : ↑(H v e μ)), ∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * Set.Nonempty.some ⋯ i = ‖h‖ ^ 2
const_out✝: ∑' (i : ℕ), t ^ 2 * (↑v i * a_g i * a_g i) = t ^ 2 * ∑' (i : ℕ), ↑v i * a_g i * a_g i
const_out: ∑' (i : ℕ), 2 * t * (↑v i * a_f i * a_g i) = 2 * t * ∑' (i : ℕ), ↑v i * a_f i * a_g i
‖f‖ ^ 2 + 2 * t * ∑' (i : ℕ), ↑v i * a_f i * a_g i + t ^ 2 * ‖g‖ ^ 2 = ‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2by
Goals accomplished! 🐙 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_f_tg:= fun i => a_f i + t * a_g i: ℕ → ℝ
a_f_tg_repr: a_f_tg ∈ set_repr (f + t * g)
distribute: ∀ (i : ℕ),
  ↑v i * a_f_tg i * a_f_tg i = ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i) + t ^ 2 * (↑v i * a_g i * a_g i)
add_summable: Summable fun i => ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i)
tsum_to_norm: ∀ (h : ↑(H v e μ)), ∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * Set.Nonempty.some ⋯ i = ‖h‖ ^ 2
const_out✝: ∑' (i : ℕ), t ^ 2 * (↑v i * a_g i * a_g i) = t ^ 2 * ∑' (i : ℕ), ↑v i * a_g i * a_g i
const_out: ∑' (i : ℕ), 2 * t * (↑v i * a_f i * a_g i) = 2 * t * ∑' (i : ℕ), ↑v i * a_f i * a_g i
∑' (i : ℕ), ↑v i * a_f i * a_g i = ⟪f, g⟫_ℝrfl
Goals accomplished! 🐙
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
‖f + t * g‖ ^ 2 = ‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_f_tg:= fun i => a_f i + t * a_g i: ℕ → ℝ
a_f_tg_repr: a_f_tg ∈ set_repr (f + t * g)
distribute: ∀ (i : ℕ),
  ↑v i * a_f_tg i * a_f_tg i = ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i) + t ^ 2 * (↑v i * a_g i * a_g i)
add_summable: Summable fun i => ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i)
tsum_to_norm: ∀ (h : ↑(H v e μ)), ∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * Set.Nonempty.some ⋯ i = ‖h‖ ^ 2
const_out✝: ∑' (i : ℕ), t ^ 2 * (↑v i * a_g i * a_g i) = t ^ 2 * ∑' (i : ℕ), ↑v i * a_g i * a_g i
const_out: ∑' (i : ℕ), 2 * t * (↑v i * a_f i * a_g i) = 2 * t * ∑' (i : ℕ), ↑v i * a_f i * a_g i
tsum_to_inner: ∑' (i : ℕ), ↑v i * a_f i * a_g i = ⟪f, g⟫_ℝ
‖f‖ ^ 2 + 2 * t * ∑' (i : ℕ), ↑v i * a_f i * a_g i + t ^ 2 * ‖g‖ ^ 2 = ‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2tsum_to_inner
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
t: ℝ
inner_nn: 0 ≤ ⟪f + t * g, f + t * g⟫_ℝ
a_f:= Set.Nonempty.some ⋯: ℕ → ℝ
a_g:= Set.Nonempty.some ⋯: ℕ → ℝ
a_f_tg:= fun i => a_f i + t * a_g i: ℕ → ℝ
a_f_tg_repr: a_f_tg ∈ set_repr (f + t * g)
distribute: ∀ (i : ℕ),
  ↑v i * a_f_tg i * a_f_tg i = ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i) + t ^ 2 * (↑v i * a_g i * a_g i)
add_summable: Summable fun i => ↑v i * a_f i * a_f i + 2 * t * (↑v i * a_f i * a_g i)
tsum_to_norm: ∀ (h : ↑(H v e μ)), ∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * Set.Nonempty.some ⋯ i = ‖h‖ ^ 2
const_out✝: ∑' (i : ℕ), t ^ 2 * (↑v i * a_g i * a_g i) = t ^ 2 * ∑' (i : ℕ), ↑v i * a_g i * a_g i
const_out: ∑' (i : ℕ), 2 * t * (↑v i * a_f i * a_g i) = 2 * t * ∑' (i : ℕ), ↑v i * a_f i * a_g i
tsum_to_inner: ∑' (i : ℕ), ↑v i * a_f i * a_g i = ⟪f, g⟫_ℝ
‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2 = ‖f‖ ^ 2 + 2 * t * ⟪f, g⟫_ℝ + t ^ 2 * ‖g‖ ^ 2]
Goals accomplished! 🐙

lemma H_cauchy_schwarz : inner f g <= ‖f‖ * ‖g‖ := by
Goals accomplished! 🐙
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
⟪f, g⟫_ℝ ≤ ‖f‖ * ‖g‖by_cases hg : ‖g‖ ≠ 0
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
pos
⟪f, g⟫_ℝ ≤ ‖f‖ * ‖g‖
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ¬‖g‖ ≠ 0
neg
⟪f, g⟫_ℝ ≤ ‖f‖ * ‖g‖
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
⟪f, g⟫_ℝ ≤ ‖f‖ * ‖g‖·
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
pos
⟪f, g⟫_ℝ ≤ ‖f‖ * ‖g‖ 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
pos
⟪f, g⟫_ℝ ≤ ‖f‖ * ‖g‖
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ¬‖g‖ ≠ 0
neg
⟪f, g⟫_ℝ ≤ ‖f‖ * ‖g‖have hg_sq := pow_ne_zero 2 hg
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
pos
⟪f, g⟫_ℝ ≤ ‖f‖ * ‖g‖
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
pos
⟪f, g⟫_ℝ ≤ ‖f‖ * ‖g‖let P := λ (t : ℝ) ↦ ‖f + t*g‖^2
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
P:= fun t => ‖f + t * g‖ ^ 2: ℝ → ℝ
pos
⟪f, g⟫_ℝ ≤ ‖f‖ * ‖g‖
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
pos
⟪f, g⟫_ℝ ≤ ‖f‖ * ‖g‖let t₀ := -inner f g / ‖g‖^2
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
P:= fun t => ‖f + t * g‖ ^ 2: ℝ → ℝ
t₀:= -⟪f, g⟫_ℝ / ‖g‖ ^ 2: ℝ
pos
⟪f, g⟫_ℝ ≤ ‖f‖ * ‖g‖

    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
pos
⟪f, g⟫_ℝ ≤ ‖f‖ * ‖g‖have P_nonneg : 0 <= P t₀  := sq_nonneg ‖f + t₀*g‖
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
P:= fun t => ‖f + t * g‖ ^ 2: ℝ → ℝ
t₀:= -⟪f, g⟫_ℝ / ‖g‖ ^ 2: ℝ
P_nonneg: 0 ≤ P t₀
pos
⟪f, g⟫_ℝ ≤ ‖f‖ * ‖g‖

    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
pos
⟪f, g⟫_ℝ ≤ ‖f‖ * ‖g‖have P_t0_val : P t₀ = ‖f‖^2 - (inner f g)^2 / ‖g‖^2 := 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
P:= fun t => ‖f + t * g‖ ^ 2: ℝ → ℝ
t₀:= -⟪f, g⟫_ℝ / ‖g‖ ^ 2: ℝ
P_nonneg: 0 ≤ P t₀
pos
⟪f, g⟫_ℝ ≤ ‖f‖ * ‖g‖by
Goals accomplished! 🐙 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
P:= fun t => ‖f + t * g‖ ^ 2: ℝ → ℝ
t₀:= -⟪f, g⟫_ℝ / ‖g‖ ^ 2: ℝ
P_nonneg: 0 ≤ P t₀
P t₀ = ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2{
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
P:= fun t => ‖f + t * g‖ ^ 2: ℝ → ℝ
t₀:= -⟪f, g⟫_ℝ / ‖g‖ ^ 2: ℝ
P_nonneg: 0 ≤ P t₀
P t₀ = ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2
      
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
P:= fun t => ‖f + t * g‖ ^ 2: ℝ → ℝ
t₀:= -⟪f, g⟫_ℝ / ‖g‖ ^ 2: ℝ
P_nonneg: 0 ≤ P t₀
P t₀ = ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2have P_t0 : P t₀ = ‖f‖^2 + (2 : ℝ) * t₀ * inner f g + t₀^2 * ‖g‖^2 := distrib_H_norm f g t₀
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
P:= fun t => ‖f + t * g‖ ^ 2: ℝ → ℝ
t₀:= -⟪f, g⟫_ℝ / ‖g‖ ^ 2: ℝ
P_nonneg: 0 ≤ P t₀
P_t0: P t₀ = ‖f‖ ^ 2 + 2 * t₀ * ⟪f, g⟫_ℝ + t₀ ^ 2 * ‖g‖ ^ 2
P t₀ = ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2
      
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
P:= fun t => ‖f + t * g‖ ^ 2: ℝ → ℝ
t₀:= -⟪f, g⟫_ℝ / ‖g‖ ^ 2: ℝ
P_nonneg: 0 ≤ P t₀
P t₀ = ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
P:= fun t => ‖f + t * g‖ ^ 2: ℝ → ℝ
t₀:= -⟪f, g⟫_ℝ / ‖g‖ ^ 2: ℝ
P_nonneg: 0 ≤ P t₀
P_t0: P t₀ = ‖f‖ ^ 2 + 2 * t₀ * ⟪f, g⟫_ℝ + t₀ ^ 2 * ‖g‖ ^ 2
P t₀ = ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2P_t0
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
P:= fun t => ‖f + t * g‖ ^ 2: ℝ → ℝ
t₀:= -⟪f, g⟫_ℝ / ‖g‖ ^ 2: ℝ
P_nonneg: 0 ≤ P t₀
P_t0: P t₀ = ‖f‖ ^ 2 + 2 * t₀ * ⟪f, g⟫_ℝ + t₀ ^ 2 * ‖g‖ ^ 2
‖f‖ ^ 2 + 2 * t₀ * ⟪f, g⟫_ℝ + t₀ ^ 2 * ‖g‖ ^ 2 = ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
P:= fun t => ‖f + t * g‖ ^ 2: ℝ → ℝ
t₀:= -⟪f, g⟫_ℝ / ‖g‖ ^ 2: ℝ
P_nonneg: 0 ≤ P t₀
P_t0: P t₀ = ‖f‖ ^ 2 + 2 * t₀ * ⟪f, g⟫_ℝ + t₀ ^ 2 * ‖g‖ ^ 2
‖f‖ ^ 2 + 2 * t₀ * ⟪f, g⟫_ℝ + t₀ ^ 2 * ‖g‖ ^ 2 = ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2
      
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
P:= fun t => ‖f + t * g‖ ^ 2: ℝ → ℝ
t₀:= -⟪f, g⟫_ℝ / ‖g‖ ^ 2: ℝ
P_nonneg: 0 ≤ P t₀
P t₀ = ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
P:= fun t => ‖f + t * g‖ ^ 2: ℝ → ℝ
t₀:= -⟪f, g⟫_ℝ / ‖g‖ ^ 2: ℝ
P_nonneg: 0 ≤ P t₀
P_t0: P t₀ = ‖f‖ ^ 2 + 2 * t₀ * ⟪f, g⟫_ℝ + t₀ ^ 2 * ‖g‖ ^ 2
‖f‖ ^ 2 + 2 * t₀ * ⟪f, g⟫_ℝ + t₀ ^ 2 * ‖g‖ ^ 2 = ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2show t₀^2 = (inner f g)^2 / (‖g‖^2 * ‖g‖^2) by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
P:= fun t => ‖f + t * g‖ ^ 2: ℝ → ℝ
t₀:= -⟪f, g⟫_ℝ / ‖g‖ ^ 2: ℝ
P_nonneg: 0 ≤ P t₀
P_t0: P t₀ = ‖f‖ ^ 2 + 2 * t₀ * ⟪f, g⟫_ℝ + t₀ ^ 2 * ‖g‖ ^ 2
‖f‖ ^ 2 + 2 * t₀ * ⟪f, g⟫_ℝ + t₀ ^ 2 * ‖g‖ ^ 2 = ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2ring
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
P:= fun t => ‖f + t * g‖ ^ 2: ℝ → ℝ
t₀:= -⟪f, g⟫_ℝ / ‖g‖ ^ 2: ℝ
P_nonneg: 0 ≤ P t₀
P_t0: P t₀ = ‖f‖ ^ 2 + 2 * t₀ * ⟪f, g⟫_ℝ + t₀ ^ 2 * ‖g‖ ^ 2
‖f‖ ^ 2 + 2 * t₀ * ⟪f, g⟫_ℝ + ⟪f, g⟫_ℝ ^ 2 / (‖g‖ ^ 2 * ‖g‖ ^ 2) * ‖g‖ ^ 2 = ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2
      
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
P:= fun t => ‖f + t * g‖ ^ 2: ℝ → ℝ
t₀:= -⟪f, g⟫_ℝ / ‖g‖ ^ 2: ℝ
P_nonneg: 0 ≤ P t₀
P t₀ = ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
P:= fun t => ‖f + t * g‖ ^ 2: ℝ → ℝ
t₀:= -⟪f, g⟫_ℝ / ‖g‖ ^ 2: ℝ
P_nonneg: 0 ≤ P t₀
P_t0: P t₀ = ‖f‖ ^ 2 + 2 * t₀ * ⟪f, g⟫_ℝ + t₀ ^ 2 * ‖g‖ ^ 2
‖f‖ ^ 2 + 2 * t₀ * ⟪f, g⟫_ℝ + ⟪f, g⟫_ℝ ^ 2 / (‖g‖ ^ 2 * ‖g‖ ^ 2) * ‖g‖ ^ 2 = ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2show (inner f g)^2 / (‖g‖^2 * ‖g‖^2) = (inner f g)^2 / ‖g‖^2 * (1:ℝ) / ‖g‖^2 by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
P:= fun t => ‖f + t * g‖ ^ 2: ℝ → ℝ
t₀:= -⟪f, g⟫_ℝ / ‖g‖ ^ 2: ℝ
P_nonneg: 0 ≤ P t₀
P_t0: P t₀ = ‖f‖ ^ 2 + 2 * t₀ * ⟪f, g⟫_ℝ + t₀ ^ 2 * ‖g‖ ^ 2
‖f‖ ^ 2 + 2 * t₀ * ⟪f, g⟫_ℝ + ⟪f, g⟫_ℝ ^ 2 / (‖g‖ ^ 2 * ‖g‖ ^ 2) * ‖g‖ ^ 2 = ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2ring
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
P:= fun t => ‖f + t * g‖ ^ 2: ℝ → ℝ
t₀:= -⟪f, g⟫_ℝ / ‖g‖ ^ 2: ℝ
P_nonneg: 0 ≤ P t₀
P_t0: P t₀ = ‖f‖ ^ 2 + 2 * t₀ * ⟪f, g⟫_ℝ + t₀ ^ 2 * ‖g‖ ^ 2
‖f‖ ^ 2 + 2 * t₀ * ⟪f, g⟫_ℝ + ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2 * 1 / ‖g‖ ^ 2 * ‖g‖ ^ 2 = ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2
      
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
P:= fun t => ‖f + t * g‖ ^ 2: ℝ → ℝ
t₀:= -⟪f, g⟫_ℝ / ‖g‖ ^ 2: ℝ
P_nonneg: 0 ≤ P t₀
P t₀ = ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
P:= fun t => ‖f + t * g‖ ^ 2: ℝ → ℝ
t₀:= -⟪f, g⟫_ℝ / ‖g‖ ^ 2: ℝ
P_nonneg: 0 ≤ P t₀
P_t0: P t₀ = ‖f‖ ^ 2 + 2 * t₀ * ⟪f, g⟫_ℝ + t₀ ^ 2 * ‖g‖ ^ 2
‖f‖ ^ 2 + 2 * t₀ * ⟪f, g⟫_ℝ + ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2 * 1 / ‖g‖ ^ 2 * ‖g‖ ^ 2 = ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2show (inner f g)^2 / ‖g‖^2 * (1:ℝ) / ‖g‖^2 * ‖g‖^2 = (inner f g)^2 / ‖g‖^2 * ((1:ℝ) / ‖g‖^2 * ‖g‖^2) by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
P:= fun t => ‖f + t * g‖ ^ 2: ℝ → ℝ
t₀:= -⟪f, g⟫_ℝ / ‖g‖ ^ 2: ℝ
P_nonneg: 0 ≤ P t₀
P_t0: P t₀ = ‖f‖ ^ 2 + 2 * t₀ * ⟪f, g⟫_ℝ + t₀ ^ 2 * ‖g‖ ^ 2
‖f‖ ^ 2 + 2 * t₀ * ⟪f, g⟫_ℝ + ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2 * 1 / ‖g‖ ^ 2 * ‖g‖ ^ 2 = ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2ring
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
P:= fun t => ‖f + t * g‖ ^ 2: ℝ → ℝ
t₀:= -⟪f, g⟫_ℝ / ‖g‖ ^ 2: ℝ
P_nonneg: 0 ≤ P t₀
P_t0: P t₀ = ‖f‖ ^ 2 + 2 * t₀ * ⟪f, g⟫_ℝ + t₀ ^ 2 * ‖g‖ ^ 2
‖f‖ ^ 2 + 2 * t₀ * ⟪f, g⟫_ℝ + ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2 * (1 / ‖g‖ ^ 2 * ‖g‖ ^ 2) = ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2
      
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
P:= fun t => ‖f + t * g‖ ^ 2: ℝ → ℝ
t₀:= -⟪f, g⟫_ℝ / ‖g‖ ^ 2: ℝ
P_nonneg: 0 ≤ P t₀
P t₀ = ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
P:= fun t => ‖f + t * g‖ ^ 2: ℝ → ℝ
t₀:= -⟪f, g⟫_ℝ / ‖g‖ ^ 2: ℝ
P_nonneg: 0 ≤ P t₀
P_t0: P t₀ = ‖f‖ ^ 2 + 2 * t₀ * ⟪f, g⟫_ℝ + t₀ ^ 2 * ‖g‖ ^ 2
‖f‖ ^ 2 + 2 * t₀ * ⟪f, g⟫_ℝ + ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2 * (1 / ‖g‖ ^ 2 * ‖g‖ ^ 2) = ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2one_div_mul_cancel hg_sq
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
P:= fun t => ‖f + t * g‖ ^ 2: ℝ → ℝ
t₀:= -⟪f, g⟫_ℝ / ‖g‖ ^ 2: ℝ
P_nonneg: 0 ≤ P t₀
P_t0: P t₀ = ‖f‖ ^ 2 + 2 * t₀ * ⟪f, g⟫_ℝ + t₀ ^ 2 * ‖g‖ ^ 2
‖f‖ ^ 2 + 2 * t₀ * ⟪f, g⟫_ℝ + ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2 * 1 = ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
P:= fun t => ‖f + t * g‖ ^ 2: ℝ → ℝ
t₀:= -⟪f, g⟫_ℝ / ‖g‖ ^ 2: ℝ
P_nonneg: 0 ≤ P t₀
P_t0: P t₀ = ‖f‖ ^ 2 + 2 * t₀ * ⟪f, g⟫_ℝ + t₀ ^ 2 * ‖g‖ ^ 2
‖f‖ ^ 2 + 2 * t₀ * ⟪f, g⟫_ℝ + ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2 * 1 = ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2
      
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
P:= fun t => ‖f + t * g‖ ^ 2: ℝ → ℝ
t₀:= -⟪f, g⟫_ℝ / ‖g‖ ^ 2: ℝ
P_nonneg: 0 ≤ P t₀
P t₀ = ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
P:= fun t => ‖f + t * g‖ ^ 2: ℝ → ℝ
t₀:= -⟪f, g⟫_ℝ / ‖g‖ ^ 2: ℝ
P_nonneg: 0 ≤ P t₀
P_t0: P t₀ = ‖f‖ ^ 2 + 2 * t₀ * ⟪f, g⟫_ℝ + t₀ ^ 2 * ‖g‖ ^ 2
‖f‖ ^ 2 + 2 * t₀ * ⟪f, g⟫_ℝ + ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2 * 1 = ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2show (inner f g)^2 / ‖g‖^2 * (1:ℝ) = (inner f g)^2 / ‖g‖^2 by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
P:= fun t => ‖f + t * g‖ ^ 2: ℝ → ℝ
t₀:= -⟪f, g⟫_ℝ / ‖g‖ ^ 2: ℝ
P_nonneg: 0 ≤ P t₀
P_t0: P t₀ = ‖f‖ ^ 2 + 2 * t₀ * ⟪f, g⟫_ℝ + t₀ ^ 2 * ‖g‖ ^ 2
‖f‖ ^ 2 + 2 * t₀ * ⟪f, g⟫_ℝ + ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2 * 1 = ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2ring
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
P:= fun t => ‖f + t * g‖ ^ 2: ℝ → ℝ
t₀:= -⟪f, g⟫_ℝ / ‖g‖ ^ 2: ℝ
P_nonneg: 0 ≤ P t₀
P_t0: P t₀ = ‖f‖ ^ 2 + 2 * t₀ * ⟪f, g⟫_ℝ + t₀ ^ 2 * ‖g‖ ^ 2
‖f‖ ^ 2 + 2 * t₀ * ⟪f, g⟫_ℝ + ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2 = ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2
      
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
P:= fun t => ‖f + t * g‖ ^ 2: ℝ → ℝ
t₀:= -⟪f, g⟫_ℝ / ‖g‖ ^ 2: ℝ
P_nonneg: 0 ≤ P t₀
P t₀ = ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
P:= fun t => ‖f + t * g‖ ^ 2: ℝ → ℝ
t₀:= -⟪f, g⟫_ℝ / ‖g‖ ^ 2: ℝ
P_nonneg: 0 ≤ P t₀
P_t0: P t₀ = ‖f‖ ^ 2 + 2 * t₀ * ⟪f, g⟫_ℝ + t₀ ^ 2 * ‖g‖ ^ 2
‖f‖ ^ 2 + 2 * t₀ * ⟪f, g⟫_ℝ + ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2 = ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2show (2:ℝ) * t₀ * inner f g = (-2:ℝ) * (inner f g)^2 / ‖g‖^2 by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
P:= fun t => ‖f + t * g‖ ^ 2: ℝ → ℝ
t₀:= -⟪f, g⟫_ℝ / ‖g‖ ^ 2: ℝ
P_nonneg: 0 ≤ P t₀
P_t0: P t₀ = ‖f‖ ^ 2 + 2 * t₀ * ⟪f, g⟫_ℝ + t₀ ^ 2 * ‖g‖ ^ 2
‖f‖ ^ 2 + 2 * t₀ * ⟪f, g⟫_ℝ + ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2 = ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2ring
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
P:= fun t => ‖f + t * g‖ ^ 2: ℝ → ℝ
t₀:= -⟪f, g⟫_ℝ / ‖g‖ ^ 2: ℝ
P_nonneg: 0 ≤ P t₀
P_t0: P t₀ = ‖f‖ ^ 2 + 2 * t₀ * ⟪f, g⟫_ℝ + t₀ ^ 2 * ‖g‖ ^ 2
‖f‖ ^ 2 + -2 * ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2 + ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2 = ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2
      
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
P:= fun t => ‖f + t * g‖ ^ 2: ℝ → ℝ
t₀:= -⟪f, g⟫_ℝ / ‖g‖ ^ 2: ℝ
P_nonneg: 0 ≤ P t₀
P t₀ = ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
P:= fun t => ‖f + t * g‖ ^ 2: ℝ → ℝ
t₀:= -⟪f, g⟫_ℝ / ‖g‖ ^ 2: ℝ
P_nonneg: 0 ≤ P t₀
P_t0: P t₀ = ‖f‖ ^ 2 + 2 * t₀ * ⟪f, g⟫_ℝ + t₀ ^ 2 * ‖g‖ ^ 2
‖f‖ ^ 2 + -2 * ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2 + ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2 = ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2show ‖f‖^2 + (-2:ℝ) * (inner f g)^2 / ‖g‖^2 + (inner f g)^2 / ‖g‖^2 = ‖f‖^2 -(inner f g)^2 / ‖g‖^2 by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
P:= fun t => ‖f + t * g‖ ^ 2: ℝ → ℝ
t₀:= -⟪f, g⟫_ℝ / ‖g‖ ^ 2: ℝ
P_nonneg: 0 ≤ P t₀
P_t0: P t₀ = ‖f‖ ^ 2 + 2 * t₀ * ⟪f, g⟫_ℝ + t₀ ^ 2 * ‖g‖ ^ 2
‖f‖ ^ 2 + -2 * ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2 + ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2 = ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2ring
Goals accomplished! 🐙]
Goals accomplished! 🐙
    }
Goals accomplished! 🐙

    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
pos
⟪f, g⟫_ℝ ≤ ‖f‖ * ‖g‖rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
P:= fun t => ‖f + t * g‖ ^ 2: ℝ → ℝ
t₀:= -⟪f, g⟫_ℝ / ‖g‖ ^ 2: ℝ
P_nonneg: 0 ≤ P t₀
P_t0_val: P t₀ = ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2
pos
⟪f, g⟫_ℝ ≤ ‖f‖ * ‖g‖P_t0_val
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
P:= fun t => ‖f + t * g‖ ^ 2: ℝ → ℝ
t₀:= -⟪f, g⟫_ℝ / ‖g‖ ^ 2: ℝ
P_nonneg: 0 ≤ ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2
P_t0_val: P t₀ = ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2
pos
⟪f, g⟫_ℝ ≤ ‖f‖ * ‖g‖]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
P:= fun t => ‖f + t * g‖ ^ 2: ℝ → ℝ
t₀:= -⟪f, g⟫_ℝ / ‖g‖ ^ 2: ℝ
P_nonneg: 0 ≤ ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2
P_t0_val: P t₀ = ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2
pos
⟪f, g⟫_ℝ ≤ ‖f‖ * ‖g‖ at P_nonneg
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
P:= fun t => ‖f + t * g‖ ^ 2: ℝ → ℝ
t₀:= -⟪f, g⟫_ℝ / ‖g‖ ^ 2: ℝ
P_nonneg: 0 ≤ ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2
P_t0_val: P t₀ = ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2
pos
⟪f, g⟫_ℝ ≤ ‖f‖ * ‖g‖

    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
pos
⟪f, g⟫_ℝ ≤ ‖f‖ * ‖g‖have sq_ineq : (inner f g)^2 <= (‖f‖ * ‖g‖)^2 := 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
P:= fun t => ‖f + t * g‖ ^ 2: ℝ → ℝ
t₀:= -⟪f, g⟫_ℝ / ‖g‖ ^ 2: ℝ
P_nonneg: 0 ≤ ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2
P_t0_val: P t₀ = ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2
pos
⟪f, g⟫_ℝ ≤ ‖f‖ * ‖g‖by
Goals accomplished! 🐙 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
P:= fun t => ‖f + t * g‖ ^ 2: ℝ → ℝ
t₀:= -⟪f, g⟫_ℝ / ‖g‖ ^ 2: ℝ
P_nonneg: 0 ≤ ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2
P_t0_val: P t₀ = ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2
⟪f, g⟫_ℝ ^ 2 ≤ (‖f‖ * ‖g‖) ^ 2{
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
P:= fun t => ‖f + t * g‖ ^ 2: ℝ → ℝ
t₀:= -⟪f, g⟫_ℝ / ‖g‖ ^ 2: ℝ
P_nonneg: 0 ≤ ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2
P_t0_val: P t₀ = ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2
⟪f, g⟫_ℝ ^ 2 ≤ (‖f‖ * ‖g‖) ^ 2
      
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
P:= fun t => ‖f + t * g‖ ^ 2: ℝ → ℝ
t₀:= -⟪f, g⟫_ℝ / ‖g‖ ^ 2: ℝ
P_nonneg: 0 ≤ ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2
P_t0_val: P t₀ = ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2
⟪f, g⟫_ℝ ^ 2 ≤ (‖f‖ * ‖g‖) ^ 2rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
P:= fun t => ‖f + t * g‖ ^ 2: ℝ → ℝ
t₀:= -⟪f, g⟫_ℝ / ‖g‖ ^ 2: ℝ
P_nonneg: 0 ≤ ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2
P_t0_val: P t₀ = ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2
⟪f, g⟫_ℝ ^ 2 ≤ (‖f‖ * ‖g‖) ^ 2show (‖f‖ * ‖g‖)^2 = ‖f‖^2 * ‖g‖^2 by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
P:= fun t => ‖f + t * g‖ ^ 2: ℝ → ℝ
t₀:= -⟪f, g⟫_ℝ / ‖g‖ ^ 2: ℝ
P_nonneg: 0 ≤ ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2
P_t0_val: P t₀ = ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2
⟪f, g⟫_ℝ ^ 2 ≤ (‖f‖ * ‖g‖) ^ 2ring
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
P:= fun t => ‖f + t * g‖ ^ 2: ℝ → ℝ
t₀:= -⟪f, g⟫_ℝ / ‖g‖ ^ 2: ℝ
P_nonneg: 0 ≤ ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2
P_t0_val: P t₀ = ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2
⟪f, g⟫_ℝ ^ 2 ≤ ‖f‖ ^ 2 * ‖g‖ ^ 2
      
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
P:= fun t => ‖f + t * g‖ ^ 2: ℝ → ℝ
t₀:= -⟪f, g⟫_ℝ / ‖g‖ ^ 2: ℝ
P_nonneg: 0 ≤ ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2
P_t0_val: P t₀ = ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2
⟪f, g⟫_ℝ ^ 2 ≤ (‖f‖ * ‖g‖) ^ 2rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
P:= fun t => ‖f + t * g‖ ^ 2: ℝ → ℝ
t₀:= -⟪f, g⟫_ℝ / ‖g‖ ^ 2: ℝ
P_nonneg: 0 ≤ ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2
P_t0_val: P t₀ = ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2
⟪f, g⟫_ℝ ^ 2 ≤ ‖f‖ ^ 2 * ‖g‖ ^ 2←(mul_inv_le_iff' (pow_two_pos_of_ne_zero hg))
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
P:= fun t => ‖f + t * g‖ ^ 2: ℝ → ℝ
t₀:= -⟪f, g⟫_ℝ / ‖g‖ ^ 2: ℝ
P_nonneg: 0 ≤ ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2
P_t0_val: P t₀ = ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2
⟪f, g⟫_ℝ ^ 2 * (‖g‖ ^ 2)⁻¹ ≤ ‖f‖ ^ 2]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
P:= fun t => ‖f + t * g‖ ^ 2: ℝ → ℝ
t₀:= -⟪f, g⟫_ℝ / ‖g‖ ^ 2: ℝ
P_nonneg: 0 ≤ ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2
P_t0_val: P t₀ = ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2
⟪f, g⟫_ℝ ^ 2 * (‖g‖ ^ 2)⁻¹ ≤ ‖f‖ ^ 2
      
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
P:= fun t => ‖f + t * g‖ ^ 2: ℝ → ℝ
t₀:= -⟪f, g⟫_ℝ / ‖g‖ ^ 2: ℝ
P_nonneg: 0 ≤ ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2
P_t0_val: P t₀ = ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2
⟪f, g⟫_ℝ ^ 2 ≤ (‖f‖ * ‖g‖) ^ 2rwa [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
P:= fun t => ‖f + t * g‖ ^ 2: ℝ → ℝ
t₀:= -⟪f, g⟫_ℝ / ‖g‖ ^ 2: ℝ
P_nonneg: 0 ≤ ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2
P_t0_val: P t₀ = ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2
⟪f, g⟫_ℝ ^ 2 * (‖g‖ ^ 2)⁻¹ ≤ ‖f‖ ^ 2←sub_nonneg
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
P:= fun t => ‖f + t * g‖ ^ 2: ℝ → ℝ
t₀:= -⟪f, g⟫_ℝ / ‖g‖ ^ 2: ℝ
P_nonneg: 0 ≤ ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2
P_t0_val: P t₀ = ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2
0 ≤ ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 * (‖g‖ ^ 2)⁻¹]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
P:= fun t => ‖f + t * g‖ ^ 2: ℝ → ℝ
t₀:= -⟪f, g⟫_ℝ / ‖g‖ ^ 2: ℝ
P_nonneg: 0 ≤ ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2
P_t0_val: P t₀ = ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2
0 ≤ ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 * (‖g‖ ^ 2)⁻¹
    }
Goals accomplished! 🐙
    --rw [←sq_abs (inner f g)] at sq_ineq
    rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
P:= fun t => ‖f + t * g‖ ^ 2: ℝ → ℝ
t₀:= -⟪f, g⟫_ℝ / ‖g‖ ^ 2: ℝ
P_nonneg: 0 ≤ ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2
P_t0_val: P t₀ = ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2
sq_ineq: ⟪f, g⟫_ℝ ^ 2 ≤ (‖f‖ * ‖g‖) ^ 2
pos
⟪f, g⟫_ℝ ≤ ‖f‖ * ‖g‖show (inner f g)^2 = (inner f g) * (inner f g) by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
P:= fun t => ‖f + t * g‖ ^ 2: ℝ → ℝ
t₀:= -⟪f, g⟫_ℝ / ‖g‖ ^ 2: ℝ
P_nonneg: 0 ≤ ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2
P_t0_val: P t₀ = ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2
sq_ineq: ⟪f, g⟫_ℝ ^ 2 ≤ (‖f‖ * ‖g‖) ^ 2
pos
⟪f, g⟫_ℝ ≤ ‖f‖ * ‖g‖ring
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
P:= fun t => ‖f + t * g‖ ^ 2: ℝ → ℝ
t₀:= -⟪f, g⟫_ℝ / ‖g‖ ^ 2: ℝ
P_nonneg: 0 ≤ ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2
P_t0_val: P t₀ = ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2
sq_ineq: ⟪f, g⟫_ℝ * ⟪f, g⟫_ℝ ≤ (‖f‖ * ‖g‖) ^ 2
pos
⟪f, g⟫_ℝ ≤ ‖f‖ * ‖g‖ at sq_ineq
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
P:= fun t => ‖f + t * g‖ ^ 2: ℝ → ℝ
t₀:= -⟪f, g⟫_ℝ / ‖g‖ ^ 2: ℝ
P_nonneg: 0 ≤ ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2
P_t0_val: P t₀ = ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2
sq_ineq: ⟪f, g⟫_ℝ * ⟪f, g⟫_ℝ ≤ (‖f‖ * ‖g‖) ^ 2
pos
⟪f, g⟫_ℝ ≤ ‖f‖ * ‖g‖
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
pos
⟪f, g⟫_ℝ ≤ ‖f‖ * ‖g‖rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
P:= fun t => ‖f + t * g‖ ^ 2: ℝ → ℝ
t₀:= -⟪f, g⟫_ℝ / ‖g‖ ^ 2: ℝ
P_nonneg: 0 ≤ ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2
P_t0_val: P t₀ = ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2
sq_ineq: ⟪f, g⟫_ℝ * ⟪f, g⟫_ℝ ≤ (‖f‖ * ‖g‖) ^ 2
pos
⟪f, g⟫_ℝ ≤ ‖f‖ * ‖g‖show (‖f‖ * ‖g‖)^2 = (‖f‖ * ‖g‖) * (‖f‖ * ‖g‖) by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
P:= fun t => ‖f + t * g‖ ^ 2: ℝ → ℝ
t₀:= -⟪f, g⟫_ℝ / ‖g‖ ^ 2: ℝ
P_nonneg: 0 ≤ ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2
P_t0_val: P t₀ = ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2
sq_ineq: ⟪f, g⟫_ℝ * ⟪f, g⟫_ℝ ≤ (‖f‖ * ‖g‖) ^ 2
pos
⟪f, g⟫_ℝ ≤ ‖f‖ * ‖g‖ring
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
P:= fun t => ‖f + t * g‖ ^ 2: ℝ → ℝ
t₀:= -⟪f, g⟫_ℝ / ‖g‖ ^ 2: ℝ
P_nonneg: 0 ≤ ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2
P_t0_val: P t₀ = ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2
sq_ineq: ⟪f, g⟫_ℝ * ⟪f, g⟫_ℝ ≤ ‖f‖ * ‖g‖ * (‖f‖ * ‖g‖)
pos
⟪f, g⟫_ℝ ≤ ‖f‖ * ‖g‖ at sq_ineq
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
P:= fun t => ‖f + t * g‖ ^ 2: ℝ → ℝ
t₀:= -⟪f, g⟫_ℝ / ‖g‖ ^ 2: ℝ
P_nonneg: 0 ≤ ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2
P_t0_val: P t₀ = ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2
sq_ineq: ⟪f, g⟫_ℝ * ⟪f, g⟫_ℝ ≤ ‖f‖ * ‖g‖ * (‖f‖ * ‖g‖)
pos
⟪f, g⟫_ℝ ≤ ‖f‖ * ‖g‖
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
pos
⟪f, g⟫_ℝ ≤ ‖f‖ * ‖g‖have norm_mul_nonneg : 0 <= ‖f‖ * ‖g‖ := Left.mul_nonneg (H_norm_nonneg f) (H_norm_nonneg g)
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
hg_sq: ‖g‖ ^ 2 ≠ 0
P:= fun t => ‖f + t * g‖ ^ 2: ℝ → ℝ
t₀:= -⟪f, g⟫_ℝ / ‖g‖ ^ 2: ℝ
P_nonneg: 0 ≤ ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2
P_t0_val: P t₀ = ‖f‖ ^ 2 - ⟪f, g⟫_ℝ ^ 2 / ‖g‖ ^ 2
sq_ineq: ⟪f, g⟫_ℝ * ⟪f, g⟫_ℝ ≤ ‖f‖ * ‖g‖ * (‖f‖ * ‖g‖)
norm_mul_nonneg: 0 ≤ ‖f‖ * ‖g‖
pos
⟪f, g⟫_ℝ ≤ ‖f‖ * ‖g‖
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ ≠ 0
pos
⟪f, g⟫_ℝ ≤ ‖f‖ * ‖g‖exact nonneg_le_nonneg_of_sq_le_sq norm_mul_nonneg sq_ineq
Goals accomplished! 🐙

  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
⟪f, g⟫_ℝ ≤ ‖f‖ * ‖g‖push_neg at hg
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ = 0
neg
⟪f, g⟫_ℝ ≤ ‖f‖ * ‖g‖
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
⟪f, g⟫_ℝ ≤ ‖f‖ * ‖g‖rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ = 0
neg
⟪f, g⟫_ℝ ≤ ‖f‖ * ‖g‖show ‖g‖ = Real.sqrt (inner g g) by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ‖g‖ = 0
neg
⟪f, g⟫_ℝ ≤ ‖f‖ * ‖g‖rfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: √⟪g, g⟫_ℝ = 0
neg
⟪f, g⟫_ℝ ≤ ‖f‖ * ‖g‖ at hg
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: √⟪g, g⟫_ℝ = 0
neg
⟪f, g⟫_ℝ ≤ ‖f‖ * ‖g‖
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
⟪f, g⟫_ℝ ≤ ‖f‖ * ‖g‖rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: √⟪g, g⟫_ℝ = 0
neg
⟪f, g⟫_ℝ ≤ ‖f‖ * ‖g‖Real.sqrt_eq_zero (inner_nonneg g)
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ⟪g, g⟫_ℝ = 0
neg
⟪f, g⟫_ℝ ≤ ‖f‖ * ‖g‖]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ⟪g, g⟫_ℝ = 0
neg
⟪f, g⟫_ℝ ≤ ‖f‖ * ‖g‖ at hg
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ⟪g, g⟫_ℝ = 0
neg
⟪f, g⟫_ℝ ≤ ‖f‖ * ‖g‖
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
⟪f, g⟫_ℝ ≤ ‖f‖ * ‖g‖have g_eq_0 := null_inner_imp_null_f g hg
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ⟪g, g⟫_ℝ = 0
g_eq_0: g = 0
neg
⟪f, g⟫_ℝ ≤ ‖f‖ * ‖g‖
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
⟪f, g⟫_ℝ ≤ ‖f‖ * ‖g‖rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ⟪g, g⟫_ℝ = 0
g_eq_0: g = 0
neg
⟪f, g⟫_ℝ ≤ ‖f‖ * ‖g‖g_eq_0
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ⟪g, g⟫_ℝ = 0
g_eq_0: g = 0
neg
⟪f, 0⟫_ℝ ≤ ‖f‖ * ‖0‖]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ⟪g, g⟫_ℝ = 0
g_eq_0: g = 0
neg
⟪f, 0⟫_ℝ ≤ ‖f‖ * ‖0‖
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
⟪f, g⟫_ℝ ≤ ‖f‖ * ‖g‖rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ⟪g, g⟫_ℝ = 0
g_eq_0: g = 0
neg
⟪f, 0⟫_ℝ ≤ ‖f‖ * ‖0‖show inner f (0 : H v e μ) = inner f 0 by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ⟪g, g⟫_ℝ = 0
g_eq_0: g = 0
neg
⟪f, 0⟫_ℝ ≤ ‖f‖ * ‖0‖rfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ⟪g, g⟫_ℝ = 0
g_eq_0: g = 0
neg
⟪f, 0⟫_ℝ ≤ ‖f‖ * ‖0‖
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
⟪f, g⟫_ℝ ≤ ‖f‖ * ‖g‖rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ⟪g, g⟫_ℝ = 0
g_eq_0: g = 0
neg
⟪f, 0⟫_ℝ ≤ ‖f‖ * ‖0‖inner_zero_eq_zero f
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ⟪g, g⟫_ℝ = 0
g_eq_0: g = 0
neg
0 ≤ ‖f‖ * ‖0‖]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ⟪g, g⟫_ℝ = 0
g_eq_0: g = 0
neg
0 ≤ ‖f‖ * ‖0‖
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
⟪f, g⟫_ℝ ≤ ‖f‖ * ‖g‖rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ⟪g, g⟫_ℝ = 0
g_eq_0: g = 0
neg
0 ≤ ‖f‖ * ‖0‖show norm (0 : H v e μ) = Real.sqrt (inner (0 : H v e μ) 0) by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ⟪g, g⟫_ℝ = 0
g_eq_0: g = 0
neg
0 ≤ ‖f‖ * ‖0‖rfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ⟪g, g⟫_ℝ = 0
g_eq_0: g = 0
neg
0 ≤ ‖f‖ * √⟪0, 0⟫_ℝ
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
⟪f, g⟫_ℝ ≤ ‖f‖ * ‖g‖rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ⟪g, g⟫_ℝ = 0
g_eq_0: g = 0
neg
0 ≤ ‖f‖ * √⟪0, 0⟫_ℝinner_zero_eq_zero 0
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ⟪g, g⟫_ℝ = 0
g_eq_0: g = 0
neg
0 ≤ ‖f‖ * √0]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hg: ⟪g, g⟫_ℝ = 0
g_eq_0: g = 0
neg
0 ≤ ‖f‖ * √0
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
⟪f, g⟫_ℝ ≤ ‖f‖ * ‖g‖simp
[Meta.Tactic.simp.rewrite] Real.sqrt_zero:1000, √0 ==> 0
[Meta.Tactic.simp.rewrite] @mul_zero:1000, ‖f‖ * 0 ==> 0
[Meta.Tactic.simp.rewrite] @le_refl:1000, 0 ≤ 0 ==> True
Goals accomplished! 🐙

lemma ineq_add_norm : ‖f + g‖ <= ‖f‖ + ‖g‖ := by
Goals accomplished! 🐙
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
‖f + g‖ ≤ ‖f‖ + ‖g‖apply nonneg_le_nonneg_of_sq_le_sq
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hb
0 ≤ ‖f‖ + ‖g‖
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
h
‖f + g‖ * ‖f + g‖ ≤ (‖f‖ + ‖g‖) * (‖f‖ + ‖g‖)
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
‖f + g‖ ≤ ‖f‖ + ‖g‖·
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hb
0 ≤ ‖f‖ + ‖g‖ 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
hb
0 ≤ ‖f‖ + ‖g‖
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
h
‖f + g‖ * ‖f + g‖ ≤ (‖f‖ + ‖g‖) * (‖f‖ + ‖g‖)exact Left.add_nonneg (H_norm_nonneg f) (H_norm_nonneg g)
Goals accomplished! 🐙
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
‖f + g‖ ≤ ‖f‖ + ‖g‖rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
h
‖f + g‖ * ‖f + g‖ ≤ (‖f‖ + ‖g‖) * (‖f‖ + ‖g‖)show ‖f + g‖ * ‖f + g‖ = ‖f + g‖^2 by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
h
‖f + g‖ * ‖f + g‖ ≤ (‖f‖ + ‖g‖) * (‖f‖ + ‖g‖)ring
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
h
‖f + g‖ ^ 2 ≤ (‖f‖ + ‖g‖) * (‖f‖ + ‖g‖)
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
‖f + g‖ ≤ ‖f‖ + ‖g‖rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
h
‖f + g‖ ^ 2 ≤ (‖f‖ + ‖g‖) * (‖f‖ + ‖g‖)show (‖f‖ + ‖g‖) * (‖f‖ + ‖g‖) = (‖f‖ + ‖g‖)^2 by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
h
‖f + g‖ ^ 2 ≤ (‖f‖ + ‖g‖) * (‖f‖ + ‖g‖)ring
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
h
‖f + g‖ ^ 2 ≤ (‖f‖ + ‖g‖) ^ 2

  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
‖f + g‖ ≤ ‖f‖ + ‖g‖have sq_ineq : ‖f‖^2 + (2 : ℝ) * inner f g + ‖g‖^2 <= (‖f‖ + ‖g‖)^2 := 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
h
‖f + g‖ ^ 2 ≤ (‖f‖ + ‖g‖) ^ 2by
Goals accomplished! 🐙 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
‖f‖ ^ 2 + 2 * ⟪f, g⟫_ℝ + ‖g‖ ^ 2 ≤ (‖f‖ + ‖g‖) ^ 2{
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
‖f‖ ^ 2 + 2 * ⟪f, g⟫_ℝ + ‖g‖ ^ 2 ≤ (‖f‖ + ‖g‖) ^ 2

    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
‖f‖ ^ 2 + 2 * ⟪f, g⟫_ℝ + ‖g‖ ^ 2 ≤ (‖f‖ + ‖g‖) ^ 2have cauchy_schwarz : 2 * inner f g <= 2 * (‖f‖ * ‖g‖) := 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
‖f‖ ^ 2 + 2 * ⟪f, g⟫_ℝ + ‖g‖ ^ 2 ≤ (‖f‖ + ‖g‖) ^ 2by
Goals accomplished! 🐙 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
2 * ⟪f, g⟫_ℝ ≤ 2 * (‖f‖ * ‖g‖){
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
2 * ⟪f, g⟫_ℝ ≤ 2 * (‖f‖ * ‖g‖)
      
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
2 * ⟪f, g⟫_ℝ ≤ 2 * (‖f‖ * ‖g‖)have := H_cauchy_schwarz f g
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
this: ⟪f, g⟫_ℝ ≤ ‖f‖ * ‖g‖
2 * ⟪f, g⟫_ℝ ≤ 2 * (‖f‖ * ‖g‖)
      
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
2 * ⟪f, g⟫_ℝ ≤ 2 * (‖f‖ * ‖g‖)linarith
Goals accomplished! 🐙
    }
Goals accomplished! 🐙

    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
‖f‖ ^ 2 + 2 * ⟪f, g⟫_ℝ + ‖g‖ ^ 2 ≤ (‖f‖ + ‖g‖) ^ 2have ineq := add_le_add_right (add_le_add_left cauchy_schwarz (‖f‖^2)) (‖g‖^2)
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
cauchy_schwarz: 2 * ⟪f, g⟫_ℝ ≤ 2 * (‖f‖ * ‖g‖)
ineq: ‖f‖ ^ 2 + 2 * ⟪f, g⟫_ℝ + ‖g‖ ^ 2 ≤ ‖f‖ ^ 2 + 2 * (‖f‖ * ‖g‖) + ‖g‖ ^ 2
‖f‖ ^ 2 + 2 * ⟪f, g⟫_ℝ + ‖g‖ ^ 2 ≤ (‖f‖ + ‖g‖) ^ 2
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
‖f‖ ^ 2 + 2 * ⟪f, g⟫_ℝ + ‖g‖ ^ 2 ≤ (‖f‖ + ‖g‖) ^ 2rwa [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
cauchy_schwarz: 2 * ⟪f, g⟫_ℝ ≤ 2 * (‖f‖ * ‖g‖)
ineq: ‖f‖ ^ 2 + 2 * ⟪f, g⟫_ℝ + ‖g‖ ^ 2 ≤ ‖f‖ ^ 2 + 2 * (‖f‖ * ‖g‖) + ‖g‖ ^ 2
‖f‖ ^ 2 + 2 * ⟪f, g⟫_ℝ + ‖g‖ ^ 2 ≤ (‖f‖ + ‖g‖) ^ 2show ‖f‖^2 + (2 : ℝ) * (‖f‖ * ‖g‖) + ‖g‖^2 = (‖f‖ + ‖g‖)^2 by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
cauchy_schwarz: 2 * ⟪f, g⟫_ℝ ≤ 2 * (‖f‖ * ‖g‖)
ineq: ‖f‖ ^ 2 + 2 * ⟪f, g⟫_ℝ + ‖g‖ ^ 2 ≤ ‖f‖ ^ 2 + 2 * (‖f‖ * ‖g‖) + ‖g‖ ^ 2
‖f‖ ^ 2 + 2 * ⟪f, g⟫_ℝ + ‖g‖ ^ 2 ≤ (‖f‖ + ‖g‖) ^ 2ring
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
cauchy_schwarz: 2 * ⟪f, g⟫_ℝ ≤ 2 * (‖f‖ * ‖g‖)
ineq: ‖f‖ ^ 2 + 2 * ⟪f, g⟫_ℝ + ‖g‖ ^ 2 ≤ (‖f‖ + ‖g‖) ^ 2
‖f‖ ^ 2 + 2 * ⟪f, g⟫_ℝ + ‖g‖ ^ 2 ≤ (‖f‖ + ‖g‖) ^ 2 at ineq
Goals accomplished! 🐙
  }
Goals accomplished! 🐙

  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
‖f + g‖ ≤ ‖f‖ + ‖g‖have distrib_norm := distrib_H_norm f g 1
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
sq_ineq: ‖f‖ ^ 2 + 2 * ⟪f, g⟫_ℝ + ‖g‖ ^ 2 ≤ (‖f‖ + ‖g‖) ^ 2
distrib_norm: ‖f + 1 * g‖ ^ 2 = ‖f‖ ^ 2 + 2 * 1 * ⟪f, g⟫_ℝ + 1 ^ 2 * ‖g‖ ^ 2
h
‖f + g‖ ^ 2 ≤ (‖f‖ + ‖g‖) ^ 2
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
‖f + g‖ ≤ ‖f‖ + ‖g‖rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
sq_ineq: ‖f‖ ^ 2 + 2 * ⟪f, g⟫_ℝ + ‖g‖ ^ 2 ≤ (‖f‖ + ‖g‖) ^ 2
distrib_norm: ‖f + 1 * g‖ ^ 2 = ‖f‖ ^ 2 + 2 * 1 * ⟪f, g⟫_ℝ + 1 ^ 2 * ‖g‖ ^ 2
h
‖f + g‖ ^ 2 ≤ (‖f‖ + ‖g‖) ^ 2show (2:ℝ) * (1:ℝ) * inner f g = 2 * inner f g by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
sq_ineq: ‖f‖ ^ 2 + 2 * ⟪f, g⟫_ℝ + ‖g‖ ^ 2 ≤ (‖f‖ + ‖g‖) ^ 2
distrib_norm: ‖f + 1 * g‖ ^ 2 = ‖f‖ ^ 2 + 2 * 1 * ⟪f, g⟫_ℝ + 1 ^ 2 * ‖g‖ ^ 2
h
‖f + g‖ ^ 2 ≤ (‖f‖ + ‖g‖) ^ 2ring
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
sq_ineq: ‖f‖ ^ 2 + 2 * ⟪f, g⟫_ℝ + ‖g‖ ^ 2 ≤ (‖f‖ + ‖g‖) ^ 2
distrib_norm: ‖f + 1 * g‖ ^ 2 = ‖f‖ ^ 2 + 2 * ⟪f, g⟫_ℝ + 1 ^ 2 * ‖g‖ ^ 2
h
‖f + g‖ ^ 2 ≤ (‖f‖ + ‖g‖) ^ 2 at distrib_norm
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
sq_ineq: ‖f‖ ^ 2 + 2 * ⟪f, g⟫_ℝ + ‖g‖ ^ 2 ≤ (‖f‖ + ‖g‖) ^ 2
distrib_norm: ‖f + 1 * g‖ ^ 2 = ‖f‖ ^ 2 + 2 * ⟪f, g⟫_ℝ + 1 ^ 2 * ‖g‖ ^ 2
h
‖f + g‖ ^ 2 ≤ (‖f‖ + ‖g‖) ^ 2
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
‖f + g‖ ≤ ‖f‖ + ‖g‖rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
sq_ineq: ‖f‖ ^ 2 + 2 * ⟪f, g⟫_ℝ + ‖g‖ ^ 2 ≤ (‖f‖ + ‖g‖) ^ 2
distrib_norm: ‖f + 1 * g‖ ^ 2 = ‖f‖ ^ 2 + 2 * ⟪f, g⟫_ℝ + 1 ^ 2 * ‖g‖ ^ 2
h
‖f + g‖ ^ 2 ≤ (‖f‖ + ‖g‖) ^ 2show (1:ℝ)^2 * ‖g‖^2 = ‖g‖^2 by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
sq_ineq: ‖f‖ ^ 2 + 2 * ⟪f, g⟫_ℝ + ‖g‖ ^ 2 ≤ (‖f‖ + ‖g‖) ^ 2
distrib_norm: ‖f + 1 * g‖ ^ 2 = ‖f‖ ^ 2 + 2 * ⟪f, g⟫_ℝ + 1 ^ 2 * ‖g‖ ^ 2
h
‖f + g‖ ^ 2 ≤ (‖f‖ + ‖g‖) ^ 2ring
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
sq_ineq: ‖f‖ ^ 2 + 2 * ⟪f, g⟫_ℝ + ‖g‖ ^ 2 ≤ (‖f‖ + ‖g‖) ^ 2
distrib_norm: ‖f + 1 * g‖ ^ 2 = ‖f‖ ^ 2 + 2 * ⟪f, g⟫_ℝ + ‖g‖ ^ 2
h
‖f + g‖ ^ 2 ≤ (‖f‖ + ‖g‖) ^ 2 at distrib_norm
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
sq_ineq: ‖f‖ ^ 2 + 2 * ⟪f, g⟫_ℝ + ‖g‖ ^ 2 ≤ (‖f‖ + ‖g‖) ^ 2
distrib_norm: ‖f + 1 * g‖ ^ 2 = ‖f‖ ^ 2 + 2 * ⟪f, g⟫_ℝ + ‖g‖ ^ 2
h
‖f + g‖ ^ 2 ≤ (‖f‖ + ‖g‖) ^ 2
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
‖f + g‖ ≤ ‖f‖ + ‖g‖rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
sq_ineq: ‖f‖ ^ 2 + 2 * ⟪f, g⟫_ℝ + ‖g‖ ^ 2 ≤ (‖f‖ + ‖g‖) ^ 2
distrib_norm: ‖f + 1 * g‖ ^ 2 = ‖f‖ ^ 2 + 2 * ⟪f, g⟫_ℝ + ‖g‖ ^ 2
h
‖f + g‖ ^ 2 ≤ (‖f‖ + ‖g‖) ^ 2←distrib_norm
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
sq_ineq: ‖f + 1 * g‖ ^ 2 ≤ (‖f‖ + ‖g‖) ^ 2
distrib_norm: ‖f + 1 * g‖ ^ 2 = ‖f‖ ^ 2 + 2 * ⟪f, g⟫_ℝ + ‖g‖ ^ 2
h
‖f + g‖ ^ 2 ≤ (‖f‖ + ‖g‖) ^ 2]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
sq_ineq: ‖f + 1 * g‖ ^ 2 ≤ (‖f‖ + ‖g‖) ^ 2
distrib_norm: ‖f + 1 * g‖ ^ 2 = ‖f‖ ^ 2 + 2 * ⟪f, g⟫_ℝ + ‖g‖ ^ 2
h
‖f + g‖ ^ 2 ≤ (‖f‖ + ‖g‖) ^ 2 at sq_ineq
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
sq_ineq: ‖f + 1 * g‖ ^ 2 ≤ (‖f‖ + ‖g‖) ^ 2
distrib_norm: ‖f + 1 * g‖ ^ 2 = ‖f‖ ^ 2 + 2 * ⟪f, g⟫_ℝ + ‖g‖ ^ 2
h
‖f + g‖ ^ 2 ≤ (‖f‖ + ‖g‖) ^ 2
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
‖f + g‖ ≤ ‖f‖ + ‖g‖have one_mul_g_eq_g : (1 : ℝ) * g = g := 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
sq_ineq: ‖f + 1 * g‖ ^ 2 ≤ (‖f‖ + ‖g‖) ^ 2
distrib_norm: ‖f + 1 * g‖ ^ 2 = ‖f‖ ^ 2 + 2 * ⟪f, g⟫_ℝ + ‖g‖ ^ 2
h
‖f + g‖ ^ 2 ≤ (‖f‖ + ‖g‖) ^ 2by
Goals accomplished! 🐙 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
sq_ineq: ‖f + 1 * g‖ ^ 2 ≤ (‖f‖ + ‖g‖) ^ 2
distrib_norm: ‖f + 1 * g‖ ^ 2 = ‖f‖ ^ 2 + 2 * ⟪f, g⟫_ℝ + ‖g‖ ^ 2
1 * g = g{
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
sq_ineq: ‖f + 1 * g‖ ^ 2 ≤ (‖f‖ + ‖g‖) ^ 2
distrib_norm: ‖f + 1 * g‖ ^ 2 = ‖f‖ ^ 2 + 2 * ⟪f, g⟫_ℝ + ‖g‖ ^ 2
1 * g = g
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
sq_ineq: ‖f + 1 * g‖ ^ 2 ≤ (‖f‖ + ‖g‖) ^ 2
distrib_norm: ‖f + 1 * g‖ ^ 2 = ‖f‖ ^ 2 + 2 * ⟪f, g⟫_ℝ + ‖g‖ ^ 2
1 * g = gext x
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
sq_ineq: ‖f + 1 * g‖ ^ 2 ≤ (‖f‖ + ‖g‖) ^ 2
distrib_norm: ‖f + 1 * g‖ ^ 2 = ‖f‖ ^ 2 + 2 * ⟪f, g⟫_ℝ + ‖g‖ ^ 2
x: ↑Ω
a.h
↑(1 * g) x = ↑g x
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
sq_ineq: ‖f + 1 * g‖ ^ 2 ≤ (‖f‖ + ‖g‖) ^ 2
distrib_norm: ‖f + 1 * g‖ ^ 2 = ‖f‖ ^ 2 + 2 * ⟪f, g⟫_ℝ + ‖g‖ ^ 2
1 * g = grw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
sq_ineq: ‖f + 1 * g‖ ^ 2 ≤ (‖f‖ + ‖g‖) ^ 2
distrib_norm: ‖f + 1 * g‖ ^ 2 = ‖f‖ ^ 2 + 2 * ⟪f, g⟫_ℝ + ‖g‖ ^ 2
x: ↑Ω
a.h
↑(1 * g) x = ↑g xshow ((1 : ℝ) * g).1 x = (1 : ℝ) * g.1 x by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
sq_ineq: ‖f + 1 * g‖ ^ 2 ≤ (‖f‖ + ‖g‖) ^ 2
distrib_norm: ‖f + 1 * g‖ ^ 2 = ‖f‖ ^ 2 + 2 * ⟪f, g⟫_ℝ + ‖g‖ ^ 2
x: ↑Ω
a.h
↑(1 * g) x = ↑g xrfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
sq_ineq: ‖f + 1 * g‖ ^ 2 ≤ (‖f‖ + ‖g‖) ^ 2
distrib_norm: ‖f + 1 * g‖ ^ 2 = ‖f‖ ^ 2 + 2 * ⟪f, g⟫_ℝ + ‖g‖ ^ 2
x: ↑Ω
a.h
1 * ↑g x = ↑g x
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
sq_ineq: ‖f + 1 * g‖ ^ 2 ≤ (‖f‖ + ‖g‖) ^ 2
distrib_norm: ‖f + 1 * g‖ ^ 2 = ‖f‖ ^ 2 + 2 * ⟪f, g⟫_ℝ + ‖g‖ ^ 2
1 * g = grw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
sq_ineq: ‖f + 1 * g‖ ^ 2 ≤ (‖f‖ + ‖g‖) ^ 2
distrib_norm: ‖f + 1 * g‖ ^ 2 = ‖f‖ ^ 2 + 2 * ⟪f, g⟫_ℝ + ‖g‖ ^ 2
x: ↑Ω
a.h
1 * ↑g x = ↑g xshow (1 : ℝ) * g.1 x = g.1 x by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
sq_ineq: ‖f + 1 * g‖ ^ 2 ≤ (‖f‖ + ‖g‖) ^ 2
distrib_norm: ‖f + 1 * g‖ ^ 2 = ‖f‖ ^ 2 + 2 * ⟪f, g⟫_ℝ + ‖g‖ ^ 2
x: ↑Ω
a.h
1 * ↑g x = ↑g xring
Goals accomplished! 🐙]
Goals accomplished! 🐙
  }
Goals accomplished! 🐙
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
‖f + g‖ ≤ ‖f‖ + ‖g‖rwa [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
sq_ineq: ‖f + 1 * g‖ ^ 2 ≤ (‖f‖ + ‖g‖) ^ 2
distrib_norm: ‖f + 1 * g‖ ^ 2 = ‖f‖ ^ 2 + 2 * ⟪f, g⟫_ℝ + ‖g‖ ^ 2
one_mul_g_eq_g: 1 * g = g
h
‖f + g‖ ^ 2 ≤ (‖f‖ + ‖g‖) ^ 2one_mul_g_eq_g
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
sq_ineq: ‖f + g‖ ^ 2 ≤ (‖f‖ + ‖g‖) ^ 2
distrib_norm: ‖f + 1 * g‖ ^ 2 = ‖f‖ ^ 2 + 2 * ⟪f, g⟫_ℝ + ‖g‖ ^ 2
one_mul_g_eq_g: 1 * g = g
h
‖f + g‖ ^ 2 ≤ (‖f‖ + ‖g‖) ^ 2]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
sq_ineq: ‖f + g‖ ^ 2 ≤ (‖f‖ + ‖g‖) ^ 2
distrib_norm: ‖f + 1 * g‖ ^ 2 = ‖f‖ ^ 2 + 2 * ⟪f, g⟫_ℝ + ‖g‖ ^ 2
one_mul_g_eq_g: 1 * g = g
h
‖f + g‖ ^ 2 ≤ (‖f‖ + ‖g‖) ^ 2 at sq_ineq
Goals accomplished! 🐙

end Inner


/-
  We show properties on the distance induced by the inner product of H.
-/

namespace Dist

open Inner Ring Group

variable {v : eigen} {e : ℕ → Ω → ℝ} {μ : Measure Ω} [IsFiniteMeasure μ] (f g : H v e μ)

lemma H_dist_self (a : H v e μ) : dist a a = 0 := by
Goals accomplished! 🐙
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
dist a a = 0rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
dist a a = 0show dist a a = norm (a - a) by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
dist a a = 0rfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
‖a - a‖ = 0
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
dist a a = 0have a_sub_a_eq_0 : a - a = 0 := 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
‖a - a‖ = 0by
Goals accomplished! 🐙 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
a - a = 0{
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
a - a = 0
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
a - a = 0rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
a - a = 0show a - a = a + (-1 : ℝ) * a by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
a - a = 0rfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
a + -1 * a = 0
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
a - a = 0ext x
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
x: ↑Ω
a.h
↑(a + -1 * a) x = ↑0 x
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
a - a = 0rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
x: ↑Ω
a.h
↑(a + -1 * a) x = ↑0 xshow (a + (-1 : ℝ) * a).1 x = a.1 x + ((-1 : ℝ) * a).1 x by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
x: ↑Ω
a.h
↑(a + -1 * a) x = ↑0 xrfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
x: ↑Ω
a.h
↑a x + ↑(-1 * a) x = ↑0 x
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
a - a = 0rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
x: ↑Ω
a.h
↑a x + ↑(-1 * a) x = ↑0 xshow ((-1 : ℝ) * a).1 x = (-1 : ℝ) * a.1 x by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
x: ↑Ω
a.h
↑a x + ↑(-1 * a) x = ↑0 xrfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
x: ↑Ω
a.h
↑a x + -1 * ↑a x = ↑0 x
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
a - a = 0rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
x: ↑Ω
a.h
↑a x + -1 * ↑a x = ↑0 xshow (0 : H v e μ).1 x = 0 by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
x: ↑Ω
a.h
↑a x + -1 * ↑a x = ↑0 xrfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
x: ↑Ω
a.h
↑a x + -1 * ↑a x = 0
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
a - a = 0ring
Goals accomplished! 🐙
  }
Goals accomplished! 🐙
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
dist a a = 0rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
a_sub_a_eq_0: a - a = 0
‖a - a‖ = 0a_sub_a_eq_0
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
a_sub_a_eq_0: a - a = 0
‖0‖ = 0]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
a_sub_a_eq_0: a - a = 0
‖0‖ = 0
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
dist a a = 0rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
a_sub_a_eq_0: a - a = 0
‖0‖ = 0show norm (0 : H v e μ) = Real.sqrt (inner (0 : H v e μ) 0) by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
a_sub_a_eq_0: a - a = 0
‖0‖ = 0rfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
a_sub_a_eq_0: a - a = 0
√⟪0, 0⟫_ℝ = 0
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
dist a a = 0rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
a_sub_a_eq_0: a - a = 0
√⟪0, 0⟫_ℝ = 0inner_zero_eq_zero
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
a_sub_a_eq_0: a - a = 0
√0 = 0]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
a_sub_a_eq_0: a - a = 0
√0 = 0
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a: ↑(H v e μ)
dist a a = 0exact Real.sqrt_zero
Goals accomplished! 🐙

lemma dist_rw (a b : H v e μ) : (dist a b) = Real.sqrt (∑' i, v.1 i * ((set_repr_ne a).some i - (set_repr_ne b).some i)^2) := by
Goals accomplished! 🐙
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
dist a b = √(∑' (i : ℕ), ↑v i * (Set.Nonempty.some ⋯ i - Set.Nonempty.some ⋯ i) ^ 2)rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
dist a b = √(∑' (i : ℕ), ↑v i * (Set.Nonempty.some ⋯ i - Set.Nonempty.some ⋯ i) ^ 2)show dist a b = norm (a - b) by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
dist a b = √(∑' (i : ℕ), ↑v i * (Set.Nonempty.some ⋯ i - Set.Nonempty.some ⋯ i) ^ 2)rfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
‖a - b‖ = √(∑' (i : ℕ), ↑v i * (Set.Nonempty.some ⋯ i - Set.Nonempty.some ⋯ i) ^ 2)
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
dist a b = √(∑' (i : ℕ), ↑v i * (Set.Nonempty.some ⋯ i - Set.Nonempty.some ⋯ i) ^ 2)rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
‖a - b‖ = √(∑' (i : ℕ), ↑v i * (Set.Nonempty.some ⋯ i - Set.Nonempty.some ⋯ i) ^ 2)show norm (a - b) = Real.sqrt (H_inner (a - b) (a - b)) by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
‖a - b‖ = √(∑' (i : ℕ), ↑v i * (Set.Nonempty.some ⋯ i - Set.Nonempty.some ⋯ i) ^ 2)rfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
√(H_inner (a - b) (a - b)) = √(∑' (i : ℕ), ↑v i * (Set.Nonempty.some ⋯ i - Set.Nonempty.some ⋯ i) ^ 2)
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
dist a b = √(∑' (i : ℕ), ↑v i * (Set.Nonempty.some ⋯ i - Set.Nonempty.some ⋯ i) ^ 2)unfold H_inner
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
√(∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * Set.Nonempty.some ⋯ i) =
  √(∑' (i : ℕ), ↑v i * (Set.Nonempty.some ⋯ i - Set.Nonempty.some ⋯ i) ^ 2)
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
dist a b = √(∑' (i : ℕ), ↑v i * (Set.Nonempty.some ⋯ i - Set.Nonempty.some ⋯ i) ^ 2)let repr := λ i ↦ (set_repr_ne a).some i + (-1 : ℝ) * (set_repr_ne b).some i
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
repr:= fun i => Set.Nonempty.some ⋯ i + -1 * Set.Nonempty.some ⋯ i: ℕ → ℝ
√(∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * Set.Nonempty.some ⋯ i) =
  √(∑' (i : ℕ), ↑v i * (Set.Nonempty.some ⋯ i - Set.Nonempty.some ⋯ i) ^ 2)
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
dist a b = √(∑' (i : ℕ), ↑v i * (Set.Nonempty.some ⋯ i - Set.Nonempty.some ⋯ i) ^ 2)have repr_a_minus_b : repr ∈ set_repr (a - b) := 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
repr:= fun i => Set.Nonempty.some ⋯ i + -1 * Set.Nonempty.some ⋯ i: ℕ → ℝ
√(∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * Set.Nonempty.some ⋯ i) =
  √(∑' (i : ℕ), ↑v i * (Set.Nonempty.some ⋯ i - Set.Nonempty.some ⋯ i) ^ 2)by
Goals accomplished! 🐙 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
repr:= fun i => Set.Nonempty.some ⋯ i + -1 * Set.Nonempty.some ⋯ i: ℕ → ℝ
repr ∈ set_repr (a - b){
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
repr:= fun i => Set.Nonempty.some ⋯ i + -1 * Set.Nonempty.some ⋯ i: ℕ → ℝ
repr ∈ set_repr (a - b)
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
repr:= fun i => Set.Nonempty.some ⋯ i + -1 * Set.Nonempty.some ⋯ i: ℕ → ℝ
repr ∈ set_repr (a - b)have repr_minus_b :
    (λ i ↦ (-1 : ℝ) * (set_repr_ne b).some i) ∈ set_repr ((-1 : ℝ) * b) :=
      ⟨mul_repr b (-1), mul_summable b (-1)⟩
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
repr:= fun i => Set.Nonempty.some ⋯ i + -1 * Set.Nonempty.some ⋯ i: ℕ → ℝ
repr_minus_b: (fun i => -1 * Set.Nonempty.some ⋯ i) ∈ set_repr (-1 * b)
repr ∈ set_repr (a - b)

    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
repr:= fun i => Set.Nonempty.some ⋯ i + -1 * Set.Nonempty.some ⋯ i: ℕ → ℝ
repr ∈ set_repr (a - b)have repr_a_sub_b :
      (λ i ↦ (set_repr_ne a).some i + (set_repr_ne ((-1 : ℝ) * b)).some i) ∈ set_repr (a + (-1 : ℝ) * b) :=
        ⟨add_repr a ((-1 : ℝ) * b), add_summable a ((-1 : ℝ) * b)⟩
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
repr:= fun i => Set.Nonempty.some ⋯ i + -1 * Set.Nonempty.some ⋯ i: ℕ → ℝ
repr_minus_b: (fun i => -1 * Set.Nonempty.some ⋯ i) ∈ set_repr (-1 * b)
repr_a_sub_b: (fun i => Set.Nonempty.some ⋯ i + Set.Nonempty.some ⋯ i) ∈ set_repr (a + -1 * b)
repr ∈ set_repr (a - b)

    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
repr:= fun i => Set.Nonempty.some ⋯ i + -1 * Set.Nonempty.some ⋯ i: ℕ → ℝ
repr ∈ set_repr (a - b)rwa [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
repr:= fun i => Set.Nonempty.some ⋯ i + -1 * Set.Nonempty.some ⋯ i: ℕ → ℝ
repr_minus_b: (fun i => -1 * Set.Nonempty.some ⋯ i) ∈ set_repr (-1 * b)
repr_a_sub_b: (fun i => Set.Nonempty.some ⋯ i + Set.Nonempty.some ⋯ i) ∈ set_repr (a + -1 * b)
repr ∈ set_repr (a - b)unique_choice repr_minus_b
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
repr:= fun i => Set.Nonempty.some ⋯ i + -1 * Set.Nonempty.some ⋯ i: ℕ → ℝ
repr_minus_b: (fun i => -1 * Set.Nonempty.some ⋯ i) ∈ set_repr (-1 * b)
repr_a_sub_b: (fun i => Set.Nonempty.some ⋯ i + (fun i => -1 * Set.Nonempty.some ⋯ i) i) ∈ set_repr (a + -1 * b)
repr ∈ set_repr (a - b)]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
repr:= fun i => Set.Nonempty.some ⋯ i + -1 * Set.Nonempty.some ⋯ i: ℕ → ℝ
repr_minus_b: (fun i => -1 * Set.Nonempty.some ⋯ i) ∈ set_repr (-1 * b)
repr_a_sub_b: (fun i => Set.Nonempty.some ⋯ i + (fun i => -1 * Set.Nonempty.some ⋯ i) i) ∈ set_repr (a + -1 * b)
repr ∈ set_repr (a - b) at repr_a_sub_b
Goals accomplished! 🐙
  }
Goals accomplished! 🐙
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
dist a b = √(∑' (i : ℕ), ↑v i * (Set.Nonempty.some ⋯ i - Set.Nonempty.some ⋯ i) ^ 2)rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
repr:= fun i => Set.Nonempty.some ⋯ i + -1 * Set.Nonempty.some ⋯ i: ℕ → ℝ
repr_a_minus_b: repr ∈ set_repr (a - b)
√(∑' (i : ℕ), ↑v i * Set.Nonempty.some ⋯ i * Set.Nonempty.some ⋯ i) =
  √(∑' (i : ℕ), ↑v i * (Set.Nonempty.some ⋯ i - Set.Nonempty.some ⋯ i) ^ 2)unique_choice repr_a_minus_b
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
repr:= fun i => Set.Nonempty.some ⋯ i + -1 * Set.Nonempty.some ⋯ i: ℕ → ℝ
repr_a_minus_b: repr ∈ set_repr (a - b)
√(∑' (i : ℕ), ↑v i * repr i * repr i) = √(∑' (i : ℕ), ↑v i * (Set.Nonempty.some ⋯ i - Set.Nonempty.some ⋯ i) ^ 2)]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
repr:= fun i => Set.Nonempty.some ⋯ i + -1 * Set.Nonempty.some ⋯ i: ℕ → ℝ
repr_a_minus_b: repr ∈ set_repr (a - b)
√(∑' (i : ℕ), ↑v i * repr i * repr i) = √(∑' (i : ℕ), ↑v i * (Set.Nonempty.some ⋯ i - Set.Nonempty.some ⋯ i) ^ 2)
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
dist a b = √(∑' (i : ℕ), ↑v i * (Set.Nonempty.some ⋯ i - Set.Nonempty.some ⋯ i) ^ 2)let a_r := (set_repr_ne a).some
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
repr:= fun i => Set.Nonempty.some ⋯ i + -1 * Set.Nonempty.some ⋯ i: ℕ → ℝ
repr_a_minus_b: repr ∈ set_repr (a - b)
a_r:= Set.Nonempty.some ⋯: ℕ → ℝ
√(∑' (i : ℕ), ↑v i * repr i * repr i) = √(∑' (i : ℕ), ↑v i * (Set.Nonempty.some ⋯ i - Set.Nonempty.some ⋯ i) ^ 2)
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
dist a b = √(∑' (i : ℕ), ↑v i * (Set.Nonempty.some ⋯ i - Set.Nonempty.some ⋯ i) ^ 2)let b_r := (set_repr_ne b).some
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
repr:= fun i => Set.Nonempty.some ⋯ i + -1 * Set.Nonempty.some ⋯ i: ℕ → ℝ
repr_a_minus_b: repr ∈ set_repr (a - b)
a_r:= Set.Nonempty.some ⋯: ℕ → ℝ
b_r:= Set.Nonempty.some ⋯: ℕ → ℝ
√(∑' (i : ℕ), ↑v i * repr i * repr i) = √(∑' (i : ℕ), ↑v i * (Set.Nonempty.some ⋯ i - Set.Nonempty.some ⋯ i) ^ 2)
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
dist a b = √(∑' (i : ℕ), ↑v i * (Set.Nonempty.some ⋯ i - Set.Nonempty.some ⋯ i) ^ 2)simp_rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
repr:= fun i => Set.Nonempty.some ⋯ i + -1 * Set.Nonempty.some ⋯ i: ℕ → ℝ
repr_a_minus_b: repr ∈ set_repr (a - b)
a_r:= Set.Nonempty.some ⋯: ℕ → ℝ
b_r:= Set.Nonempty.some ⋯: ℕ → ℝ
√(∑' (i : ℕ), ↑v i * repr i * repr i) = √(∑' (i : ℕ), ↑v i * (Set.Nonempty.some ⋯ i - Set.Nonempty.some ⋯ i) ^ 2)show ∀ i, v.1 i * repr i * repr i = v.1 i * (repr i)^2 by 
[Meta.Tactic.simp.rewrite] show ∀ i, v.1 i * repr i * repr i = v.1 i * (repr i) ^ 2 by intro i;
        ring:1000, ↑v i * repr i * repr i ==> ↑v i * repr i ^ 2
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
repr:= fun i => Set.Nonempty.some ⋯ i + -1 * Set.Nonempty.some ⋯ i: ℕ → ℝ
repr_a_minus_b: repr ∈ set_repr (a - b)
a_r:= Set.Nonempty.some ⋯: ℕ → ℝ
b_r:= Set.Nonempty.some ⋯: ℕ → ℝ
√(∑' (i : ℕ), ↑v i * repr i * repr i) = √(∑' (i : ℕ), ↑v i * (Set.Nonempty.some ⋯ i - Set.Nonempty.some ⋯ i) ^ 2)intro i
[Meta.Tactic.simp.rewrite] show ∀ i, v.1 i * repr i * repr i = v.1 i * (repr i) ^ 2 by intro i;
        ring:1000, ↑v i * repr i * repr i ==> ↑v i * repr i ^ 2
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
repr:= fun i => Set.Nonempty.some ⋯ i + -1 * Set.Nonempty.some ⋯ i: ℕ → ℝ
repr_a_minus_b: repr ∈ set_repr (a - b)
a_r:= Set.Nonempty.some ⋯: ℕ → ℝ
b_r:= Set.Nonempty.some ⋯: ℕ → ℝ
i: ℕ
↑v i * repr i * repr i = ↑v i * repr i ^ 2;
[Meta.Tactic.simp.rewrite] show ∀ i, v.1 i * repr i * repr i = v.1 i * (repr i) ^ 2 by intro i;
        ring:1000, ↑v i * repr i * repr i ==> ↑v i * repr i ^ 2
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
repr:= fun i => Set.Nonempty.some ⋯ i + -1 * Set.Nonempty.some ⋯ i: ℕ → ℝ
repr_a_minus_b: repr ∈ set_repr (a - b)
a_r:= Set.Nonempty.some ⋯: ℕ → ℝ
b_r:= Set.Nonempty.some ⋯: ℕ → ℝ
i: ℕ
↑v i * repr i * repr i = ↑v i * repr i ^ 2 
[Meta.Tactic.simp.rewrite] show ∀ i, v.1 i * repr i * repr i = v.1 i * (repr i) ^ 2 by intro i;
        ring:1000, ↑v i * repr i * repr i ==> ↑v i * repr i ^ 2
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
repr:= fun i => Set.Nonempty.some ⋯ i + -1 * Set.Nonempty.some ⋯ i: ℕ → ℝ
repr_a_minus_b: repr ∈ set_repr (a - b)
a_r:= Set.Nonempty.some ⋯: ℕ → ℝ
b_r:= Set.Nonempty.some ⋯: ℕ → ℝ
∀ (i : ℕ), ↑v i * repr i * repr i = ↑v i * repr i ^ 2ring
[Meta.Tactic.simp.rewrite] show ∀ i, v.1 i * repr i * repr i = v.1 i * (repr i) ^ 2 by intro i;
        ring:1000, ↑v i * repr i * repr i ==> ↑v i * repr i ^ 2
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
repr:= fun i => Set.Nonempty.some ⋯ i + -1 * Set.Nonempty.some ⋯ i: ℕ → ℝ
repr_a_minus_b: repr ∈ set_repr (a - b)
a_r:= Set.Nonempty.some ⋯: ℕ → ℝ
b_r:= Set.Nonempty.some ⋯: ℕ → ℝ
√(∑' (i : ℕ), ↑v i * repr i ^ 2) = √(∑' (i : ℕ), ↑v i * (Set.Nonempty.some ⋯ i - Set.Nonempty.some ⋯ i) ^ 2)
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
dist a b = √(∑' (i : ℕ), ↑v i * (Set.Nonempty.some ⋯ i - Set.Nonempty.some ⋯ i) ^ 2)simp_rw[
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
repr:= fun i => Set.Nonempty.some ⋯ i + -1 * Set.Nonempty.some ⋯ i: ℕ → ℝ
repr_a_minus_b: repr ∈ set_repr (a - b)
a_r:= Set.Nonempty.some ⋯: ℕ → ℝ
b_r:= Set.Nonempty.some ⋯: ℕ → ℝ
√(∑' (i : ℕ), ↑v i * repr i ^ 2) = √(∑' (i : ℕ), ↑v i * (Set.Nonempty.some ⋯ i - Set.Nonempty.some ⋯ i) ^ 2)show ∀ i, v.1 i * (repr i)^2 = v.1 i * (a_r i - b_r i)^2 by 
[Meta.Tactic.simp.rewrite] show ∀ i, v.1 i * (repr i) ^ 2 = v.1 i * (a_r i - b_r i) ^ 2 by intro i;
        ring:1000, ↑v i * repr i ^ 2 ==> ↑v i * (a_r i - b_r i) ^ 2
[Meta.Tactic.simp.rewrite] @eq_self:1000, √(∑' (i : ℕ), ↑v i * (a_r i - b_r i) ^ 2) =
      √(∑' (i : ℕ), ↑v i * (Set.Nonempty.some ⋯ i - Set.Nonempty.some ⋯ i) ^ 2) ==> True
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
repr:= fun i => Set.Nonempty.some ⋯ i + -1 * Set.Nonempty.some ⋯ i: ℕ → ℝ
repr_a_minus_b: repr ∈ set_repr (a - b)
a_r:= Set.Nonempty.some ⋯: ℕ → ℝ
b_r:= Set.Nonempty.some ⋯: ℕ → ℝ
√(∑' (i : ℕ), ↑v i * repr i ^ 2) = √(∑' (i : ℕ), ↑v i * (Set.Nonempty.some ⋯ i - Set.Nonempty.some ⋯ i) ^ 2)intro i
[Meta.Tactic.simp.rewrite] show ∀ i, v.1 i * (repr i) ^ 2 = v.1 i * (a_r i - b_r i) ^ 2 by intro i;
        ring:1000, ↑v i * repr i ^ 2 ==> ↑v i * (a_r i - b_r i) ^ 2
[Meta.Tactic.simp.rewrite] @eq_self:1000, √(∑' (i : ℕ), ↑v i * (a_r i - b_r i) ^ 2) =
      √(∑' (i : ℕ), ↑v i * (Set.Nonempty.some ⋯ i - Set.Nonempty.some ⋯ i) ^ 2) ==> True
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
repr:= fun i => Set.Nonempty.some ⋯ i + -1 * Set.Nonempty.some ⋯ i: ℕ → ℝ
repr_a_minus_b: repr ∈ set_repr (a - b)
a_r:= Set.Nonempty.some ⋯: ℕ → ℝ
b_r:= Set.Nonempty.some ⋯: ℕ → ℝ
i: ℕ
↑v i * repr i ^ 2 = ↑v i * (a_r i - b_r i) ^ 2;
[Meta.Tactic.simp.rewrite] show ∀ i, v.1 i * (repr i) ^ 2 = v.1 i * (a_r i - b_r i) ^ 2 by intro i;
        ring:1000, ↑v i * repr i ^ 2 ==> ↑v i * (a_r i - b_r i) ^ 2
[Meta.Tactic.simp.rewrite] @eq_self:1000, √(∑' (i : ℕ), ↑v i * (a_r i - b_r i) ^ 2) =
      √(∑' (i : ℕ), ↑v i * (Set.Nonempty.some ⋯ i - Set.Nonempty.some ⋯ i) ^ 2) ==> True
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
repr:= fun i => Set.Nonempty.some ⋯ i + -1 * Set.Nonempty.some ⋯ i: ℕ → ℝ
repr_a_minus_b: repr ∈ set_repr (a - b)
a_r:= Set.Nonempty.some ⋯: ℕ → ℝ
b_r:= Set.Nonempty.some ⋯: ℕ → ℝ
i: ℕ
↑v i * repr i ^ 2 = ↑v i * (a_r i - b_r i) ^ 2 
[Meta.Tactic.simp.rewrite] show ∀ i, v.1 i * (repr i) ^ 2 = v.1 i * (a_r i - b_r i) ^ 2 by intro i;
        ring:1000, ↑v i * repr i ^ 2 ==> ↑v i * (a_r i - b_r i) ^ 2
[Meta.Tactic.simp.rewrite] @eq_self:1000, √(∑' (i : ℕ), ↑v i * (a_r i - b_r i) ^ 2) =
      √(∑' (i : ℕ), ↑v i * (Set.Nonempty.some ⋯ i - Set.Nonempty.some ⋯ i) ^ 2) ==> True
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
repr:= fun i => Set.Nonempty.some ⋯ i + -1 * Set.Nonempty.some ⋯ i: ℕ → ℝ
repr_a_minus_b: repr ∈ set_repr (a - b)
a_r:= Set.Nonempty.some ⋯: ℕ → ℝ
b_r:= Set.Nonempty.some ⋯: ℕ → ℝ
∀ (i : ℕ), ↑v i * repr i ^ 2 = ↑v i * (a_r i - b_r i) ^ 2ring
[Meta.Tactic.simp.rewrite] show ∀ i, v.1 i * (repr i) ^ 2 = v.1 i * (a_r i - b_r i) ^ 2 by intro i;
        ring:1000, ↑v i * repr i ^ 2 ==> ↑v i * (a_r i - b_r i) ^ 2
[Meta.Tactic.simp.rewrite] @eq_self:1000, √(∑' (i : ℕ), ↑v i * (a_r i - b_r i) ^ 2) =
      √(∑' (i : ℕ), ↑v i * (Set.Nonempty.some ⋯ i - Set.Nonempty.some ⋯ i) ^ 2) ==> True
Goals accomplished! 🐙]
Goals accomplished! 🐙

lemma H_dist_comm (a b : H v e μ) : dist a b = dist b a := by
Goals accomplished! 🐙
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
dist a b = dist b arw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
dist a b = dist b adist_rw a b,
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
√(∑' (i : ℕ), ↑v i * (Set.Nonempty.some ⋯ i - Set.Nonempty.some ⋯ i) ^ 2) = dist b a 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
dist a b = dist b adist_rw b a
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
√(∑' (i : ℕ), ↑v i * (Set.Nonempty.some ⋯ i - Set.Nonempty.some ⋯ i) ^ 2) =
  √(∑' (i : ℕ), ↑v i * (Set.Nonempty.some ⋯ i - Set.Nonempty.some ⋯ i) ^ 2)]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
√(∑' (i : ℕ), ↑v i * (Set.Nonempty.some ⋯ i - Set.Nonempty.some ⋯ i) ^ 2) =
  √(∑' (i : ℕ), ↑v i * (Set.Nonempty.some ⋯ i - Set.Nonempty.some ⋯ i) ^ 2)
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
dist a b = dist b alet a_r := (set_repr_ne a).some
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
a_r:= Set.Nonempty.some ⋯: ℕ → ℝ
√(∑' (i : ℕ), ↑v i * (Set.Nonempty.some ⋯ i - Set.Nonempty.some ⋯ i) ^ 2) =
  √(∑' (i : ℕ), ↑v i * (Set.Nonempty.some ⋯ i - Set.Nonempty.some ⋯ i) ^ 2)
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
dist a b = dist b alet b_r := (set_repr_ne b).some
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
a_r:= Set.Nonempty.some ⋯: ℕ → ℝ
b_r:= Set.Nonempty.some ⋯: ℕ → ℝ
√(∑' (i : ℕ), ↑v i * (Set.Nonempty.some ⋯ i - Set.Nonempty.some ⋯ i) ^ 2) =
  √(∑' (i : ℕ), ↑v i * (Set.Nonempty.some ⋯ i - Set.Nonempty.some ⋯ i) ^ 2)
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
dist a b = dist b asimp_rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
a_r:= Set.Nonempty.some ⋯: ℕ → ℝ
b_r:= Set.Nonempty.some ⋯: ℕ → ℝ
√(∑' (i : ℕ), ↑v i * (Set.Nonempty.some ⋯ i - Set.Nonempty.some ⋯ i) ^ 2) =
  √(∑' (i : ℕ), ↑v i * (Set.Nonempty.some ⋯ i - Set.Nonempty.some ⋯ i) ^ 2)show ∀ i, v.1 i * (a_r i - b_r i)^2 = v.1 i * (b_r i - a_r i)^2 by 
[Meta.Tactic.simp.rewrite] show ∀ i, v.1 i * (a_r i - b_r i) ^ 2 = v.1 i * (b_r i - a_r i) ^ 2 by intro i;
        ring:1000, ↑v i * (Set.Nonempty.some ⋯ i - Set.Nonempty.some ⋯ i) ^ 2 ==> ↑v i * (b_r i - a_r i) ^ 2
[Meta.Tactic.simp.rewrite] @eq_self:1000, √(∑' (i : ℕ), ↑v i * (b_r i - a_r i) ^ 2) =
      √(∑' (i : ℕ), ↑v i * (Set.Nonempty.some ⋯ i - Set.Nonempty.some ⋯ i) ^ 2) ==> True
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
a_r:= Set.Nonempty.some ⋯: ℕ → ℝ
b_r:= Set.Nonempty.some ⋯: ℕ → ℝ
√(∑' (i : ℕ), ↑v i * (Set.Nonempty.some ⋯ i - Set.Nonempty.some ⋯ i) ^ 2) =
  √(∑' (i : ℕ), ↑v i * (Set.Nonempty.some ⋯ i - Set.Nonempty.some ⋯ i) ^ 2)intro i
[Meta.Tactic.simp.rewrite] show ∀ i, v.1 i * (a_r i - b_r i) ^ 2 = v.1 i * (b_r i - a_r i) ^ 2 by intro i;
        ring:1000, ↑v i * (Set.Nonempty.some ⋯ i - Set.Nonempty.some ⋯ i) ^ 2 ==> ↑v i * (b_r i - a_r i) ^ 2
[Meta.Tactic.simp.rewrite] @eq_self:1000, √(∑' (i : ℕ), ↑v i * (b_r i - a_r i) ^ 2) =
      √(∑' (i : ℕ), ↑v i * (Set.Nonempty.some ⋯ i - Set.Nonempty.some ⋯ i) ^ 2) ==> True
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
a_r:= Set.Nonempty.some ⋯: ℕ → ℝ
b_r:= Set.Nonempty.some ⋯: ℕ → ℝ
i: ℕ
↑v i * (a_r i - b_r i) ^ 2 = ↑v i * (b_r i - a_r i) ^ 2;
[Meta.Tactic.simp.rewrite] show ∀ i, v.1 i * (a_r i - b_r i) ^ 2 = v.1 i * (b_r i - a_r i) ^ 2 by intro i;
        ring:1000, ↑v i * (Set.Nonempty.some ⋯ i - Set.Nonempty.some ⋯ i) ^ 2 ==> ↑v i * (b_r i - a_r i) ^ 2
[Meta.Tactic.simp.rewrite] @eq_self:1000, √(∑' (i : ℕ), ↑v i * (b_r i - a_r i) ^ 2) =
      √(∑' (i : ℕ), ↑v i * (Set.Nonempty.some ⋯ i - Set.Nonempty.some ⋯ i) ^ 2) ==> True
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
a_r:= Set.Nonempty.some ⋯: ℕ → ℝ
b_r:= Set.Nonempty.some ⋯: ℕ → ℝ
i: ℕ
↑v i * (a_r i - b_r i) ^ 2 = ↑v i * (b_r i - a_r i) ^ 2 
[Meta.Tactic.simp.rewrite] show ∀ i, v.1 i * (a_r i - b_r i) ^ 2 = v.1 i * (b_r i - a_r i) ^ 2 by intro i;
        ring:1000, ↑v i * (Set.Nonempty.some ⋯ i - Set.Nonempty.some ⋯ i) ^ 2 ==> ↑v i * (b_r i - a_r i) ^ 2
[Meta.Tactic.simp.rewrite] @eq_self:1000, √(∑' (i : ℕ), ↑v i * (b_r i - a_r i) ^ 2) =
      √(∑' (i : ℕ), ↑v i * (Set.Nonempty.some ⋯ i - Set.Nonempty.some ⋯ i) ^ 2) ==> True
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
a_r:= Set.Nonempty.some ⋯: ℕ → ℝ
b_r:= Set.Nonempty.some ⋯: ℕ → ℝ
∀ (i : ℕ), ↑v i * (a_r i - b_r i) ^ 2 = ↑v i * (b_r i - a_r i) ^ 2ring
[Meta.Tactic.simp.rewrite] show ∀ i, v.1 i * (a_r i - b_r i) ^ 2 = v.1 i * (b_r i - a_r i) ^ 2 by intro i;
        ring:1000, ↑v i * (Set.Nonempty.some ⋯ i - Set.Nonempty.some ⋯ i) ^ 2 ==> ↑v i * (b_r i - a_r i) ^ 2
[Meta.Tactic.simp.rewrite] @eq_self:1000, √(∑' (i : ℕ), ↑v i * (b_r i - a_r i) ^ 2) =
      √(∑' (i : ℕ), ↑v i * (Set.Nonempty.some ⋯ i - Set.Nonempty.some ⋯ i) ^ 2) ==> True
Goals accomplished! 🐙]
Goals accomplished! 🐙

lemma H_dist_nonneg (a b : H v e μ) : 0 <= dist a b := by
Goals accomplished! 🐙
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
0 ≤ dist a brw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
0 ≤ dist a bshow dist a b = norm (a - b) by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
0 ≤ dist a brfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
0 ≤ ‖a - b‖
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
0 ≤ dist a brw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
0 ≤ ‖a - b‖show norm (a - b) = Real.sqrt (H_inner (a - b) (a - b)) by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
0 ≤ ‖a - b‖rfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
0 ≤ √(H_inner (a - b) (a - b))
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
0 ≤ dist a bexact Real.sqrt_nonneg (H_inner (a - b) (a - b))
Goals accomplished! 🐙

lemma H_eq_of_dist_eq_zero {a b : H v e μ} : dist a b = 0 → a = b := by
Goals accomplished! 🐙
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
dist a b = 0 → a = bintro zero_dist
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
zero_dist: dist a b = 0
a = b
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
dist a b = 0 → a = brw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
zero_dist: dist a b = 0
a = bshow dist a b = norm (a - b) by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
zero_dist: dist a b = 0
a = brfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
zero_dist: ‖a - b‖ = 0
a = b at zero_dist
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
zero_dist: ‖a - b‖ = 0
a = b
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
dist a b = 0 → a = brw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
zero_dist: ‖a - b‖ = 0
a = bshow norm (a - b) = Real.sqrt (inner (a - b) (a - b)) by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
zero_dist: ‖a - b‖ = 0
a = brfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
zero_dist: √⟪a - b, a - b⟫_ℝ = 0
a = b at zero_dist
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
zero_dist: √⟪a - b, a - b⟫_ℝ = 0
a = b
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
dist a b = 0 → a = brw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
zero_dist: √⟪a - b, a - b⟫_ℝ = 0
a = bReal.sqrt_eq_zero (inner_nonneg (a - b))
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
zero_dist: ⟪a - b, a - b⟫_ℝ = 0
a = b]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
zero_dist: ⟪a - b, a - b⟫_ℝ = 0
a = b at zero_dist
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
zero_dist: ⟪a - b, a - b⟫_ℝ = 0
a = b
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
dist a b = 0 → a = bhave a_minus_b_eq_zero : a - b = 0 := null_inner_imp_null_f (a - b) zero_dist
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
zero_dist: ⟪a - b, a - b⟫_ℝ = 0
a_minus_b_eq_zero: a - b = 0
a = b
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
dist a b = 0 → a = bext x
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
zero_dist: ⟪a - b, a - b⟫_ℝ = 0
a_minus_b_eq_zero: a - b = 0
x: ↑Ω
a.h
↑a x = ↑b x
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
dist a b = 0 → a = bhave dif_imp_eq : a.1 x - b.1 x = 0 → a.1 x = b.1 x := 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
zero_dist: ⟪a - b, a - b⟫_ℝ = 0
a_minus_b_eq_zero: a - b = 0
x: ↑Ω
a.h
↑a x = ↑b xby
Goals accomplished! 🐙 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
zero_dist: ⟪a - b, a - b⟫_ℝ = 0
a_minus_b_eq_zero: a - b = 0
x: ↑Ω
↑a x - ↑b x = 0 → ↑a x = ↑b xintro h
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
zero_dist: ⟪a - b, a - b⟫_ℝ = 0
a_minus_b_eq_zero: a - b = 0
x: ↑Ω
h: ↑a x - ↑b x = 0
↑a x = ↑b x;
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
zero_dist: ⟪a - b, a - b⟫_ℝ = 0
a_minus_b_eq_zero: a - b = 0
x: ↑Ω
h: ↑a x - ↑b x = 0
↑a x = ↑b x 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
zero_dist: ⟪a - b, a - b⟫_ℝ = 0
a_minus_b_eq_zero: a - b = 0
x: ↑Ω
↑a x - ↑b x = 0 → ↑a x = ↑b xlinarith
Goals accomplished! 🐙
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
dist a b = 0 → a = bapply dif_imp_eq
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
zero_dist: ⟪a - b, a - b⟫_ℝ = 0
a_minus_b_eq_zero: a - b = 0
x: ↑Ω
dif_imp_eq: ↑a x - ↑b x = 0 → ↑a x = ↑b x
a.h
↑a x - ↑b x = 0
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
dist a b = 0 → a = brw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
zero_dist: ⟪a - b, a - b⟫_ℝ = 0
a_minus_b_eq_zero: a - b = 0
x: ↑Ω
dif_imp_eq: ↑a x - ↑b x = 0 → ↑a x = ↑b x
a.h
↑a x - ↑b x = 0show a.1 x - b.1 x = a.1 x + (-1 : ℝ) * b.1 x by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
zero_dist: ⟪a - b, a - b⟫_ℝ = 0
a_minus_b_eq_zero: a - b = 0
x: ↑Ω
dif_imp_eq: ↑a x - ↑b x = 0 → ↑a x = ↑b x
a.h
↑a x - ↑b x = 0ring
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
zero_dist: ⟪a - b, a - b⟫_ℝ = 0
a_minus_b_eq_zero: a - b = 0
x: ↑Ω
dif_imp_eq: ↑a x - ↑b x = 0 → ↑a x = ↑b x
a.h
↑a x + -1 * ↑b x = 0
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
dist a b = 0 → a = brw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
zero_dist: ⟪a - b, a - b⟫_ℝ = 0
a_minus_b_eq_zero: a - b = 0
x: ↑Ω
dif_imp_eq: ↑a x - ↑b x = 0 → ↑a x = ↑b x
a.h
↑a x + -1 * ↑b x = 0show (a - b) = a + (-1 : ℝ) * b by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
zero_dist: ⟪a - b, a - b⟫_ℝ = 0
a_minus_b_eq_zero: a - b = 0
x: ↑Ω
dif_imp_eq: ↑a x - ↑b x = 0 → ↑a x = ↑b x
a.h
↑a x + -1 * ↑b x = 0rfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
zero_dist: ⟪a - b, a - b⟫_ℝ = 0
a_minus_b_eq_zero: a + -1 * b = 0
x: ↑Ω
dif_imp_eq: ↑a x - ↑b x = 0 → ↑a x = ↑b x
a.h
↑a x + -1 * ↑b x = 0 at a_minus_b_eq_zero
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
zero_dist: ⟪a - b, a - b⟫_ℝ = 0
a_minus_b_eq_zero: a + -1 * b = 0
x: ↑Ω
dif_imp_eq: ↑a x - ↑b x = 0 → ↑a x = ↑b x
a.h
↑a x + -1 * ↑b x = 0
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
dist a b = 0 → a = bhave a_minus_b_eq_zero_val : (a + (-1 : ℝ) * b).1 = (0 : H v e μ).1 := congrArg Subtype.val a_minus_b_eq_zero
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
zero_dist: ⟪a - b, a - b⟫_ℝ = 0
a_minus_b_eq_zero: a + -1 * b = 0
x: ↑Ω
dif_imp_eq: ↑a x - ↑b x = 0 → ↑a x = ↑b x
a_minus_b_eq_zero_val: ↑(a + -1 * b) = ↑0
a.h
↑a x + -1 * ↑b x = 0
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
dist a b = 0 → a = brw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
zero_dist: ⟪a - b, a - b⟫_ℝ = 0
a_minus_b_eq_zero: a + -1 * b = 0
x: ↑Ω
dif_imp_eq: ↑a x - ↑b x = 0 → ↑a x = ↑b x
a_minus_b_eq_zero_val: ↑(a + -1 * b) = ↑0
a.h
↑a x + -1 * ↑b x = 0show (a + (-1 : ℝ) * b).1 = (λ x ↦ a.1 x + (-1 : ℝ) * b.1 x) by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
zero_dist: ⟪a - b, a - b⟫_ℝ = 0
a_minus_b_eq_zero: a + -1 * b = 0
x: ↑Ω
dif_imp_eq: ↑a x - ↑b x = 0 → ↑a x = ↑b x
a_minus_b_eq_zero_val: ↑(a + -1 * b) = ↑0
a.h
↑a x + -1 * ↑b x = 0rfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
zero_dist: ⟪a - b, a - b⟫_ℝ = 0
a_minus_b_eq_zero: a + -1 * b = 0
x: ↑Ω
dif_imp_eq: ↑a x - ↑b x = 0 → ↑a x = ↑b x
a_minus_b_eq_zero_val: (fun x => ↑a x + -1 * ↑b x) = ↑0
a.h
↑a x + -1 * ↑b x = 0 at a_minus_b_eq_zero_val
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
zero_dist: ⟪a - b, a - b⟫_ℝ = 0
a_minus_b_eq_zero: a + -1 * b = 0
x: ↑Ω
dif_imp_eq: ↑a x - ↑b x = 0 → ↑a x = ↑b x
a_minus_b_eq_zero_val: (fun x => ↑a x + -1 * ↑b x) = ↑0
a.h
↑a x + -1 * ↑b x = 0
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
dist a b = 0 → a = brw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
zero_dist: ⟪a - b, a - b⟫_ℝ = 0
a_minus_b_eq_zero: a + -1 * b = 0
x: ↑Ω
dif_imp_eq: ↑a x - ↑b x = 0 → ↑a x = ↑b x
a_minus_b_eq_zero_val: (fun x => ↑a x + -1 * ↑b x) = ↑0
a.h
↑a x + -1 * ↑b x = 0congrFun a_minus_b_eq_zero_val x
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
zero_dist: ⟪a - b, a - b⟫_ℝ = 0
a_minus_b_eq_zero: a + -1 * b = 0
x: ↑Ω
dif_imp_eq: ↑a x - ↑b x = 0 → ↑a x = ↑b x
a_minus_b_eq_zero_val: (fun x => ↑a x + -1 * ↑b x) = ↑0
a.h
↑0 x = 0]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
zero_dist: ⟪a - b, a - b⟫_ℝ = 0
a_minus_b_eq_zero: a + -1 * b = 0
x: ↑Ω
dif_imp_eq: ↑a x - ↑b x = 0 → ↑a x = ↑b x
a_minus_b_eq_zero_val: (fun x => ↑a x + -1 * ↑b x) = ↑0
a.h
↑0 x = 0
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: ↑(H v e μ)
dist a b = 0 → a = brfl
Goals accomplished! 🐙

lemma H_dist_triangle (a b c : H v e μ) : dist a c ≤ dist a b + dist b c := by
Goals accomplished! 🐙
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: ↑(H v e μ)
dist a c ≤ dist a b + dist b crw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: ↑(H v e μ)
dist a c ≤ dist a b + dist b cshow dist a c = norm (a - c) by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: ↑(H v e μ)
dist a c ≤ dist a b + dist b crfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: ↑(H v e μ)
‖a - c‖ ≤ dist a b + dist b c
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: ↑(H v e μ)
dist a c ≤ dist a b + dist b crw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: ↑(H v e μ)
‖a - c‖ ≤ dist a b + dist b cshow dist a b = norm (a - b) by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: ↑(H v e μ)
‖a - c‖ ≤ dist a b + dist b crfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: ↑(H v e μ)
‖a - c‖ ≤ ‖a - b‖ + dist b c
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: ↑(H v e μ)
dist a c ≤ dist a b + dist b crw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: ↑(H v e μ)
‖a - c‖ ≤ ‖a - b‖ + dist b cshow dist b c = norm (b - c) by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: ↑(H v e μ)
‖a - c‖ ≤ ‖a - b‖ + dist b crfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: ↑(H v e μ)
‖a - c‖ ≤ ‖a - b‖ + ‖b - c‖
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: ↑(H v e μ)
dist a c ≤ dist a b + dist b chave split_a_sub_c : a - c = (a - b) + (b - c) := 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: ↑(H v e μ)
‖a - c‖ ≤ ‖a - b‖ + ‖b - c‖by
Goals accomplished! 🐙 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: ↑(H v e μ)
a - c = a - b + (b - c){
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: ↑(H v e μ)
a - c = a - b + (b - c)
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: ↑(H v e μ)
a - c = a - b + (b - c)ext x
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: ↑(H v e μ)
x: ↑Ω
a.h
↑(a - c) x = ↑(a - b + (b - c)) x
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: ↑(H v e μ)
a - c = a - b + (b - c)rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: ↑(H v e μ)
x: ↑Ω
a.h
↑(a - c) x = ↑(a - b + (b - c)) xshow (a - b + (b - c)).1 x = (a - b).1 x + (b - c).1 x by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: ↑(H v e μ)
x: ↑Ω
a.h
↑(a - c) x = ↑(a - b + (b - c)) xrfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: ↑(H v e μ)
x: ↑Ω
a.h
↑(a - c) x = ↑(a - b) x + ↑(b - c) x
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: ↑(H v e μ)
a - c = a - b + (b - c)rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: ↑(H v e μ)
x: ↑Ω
a.h
↑(a - c) x = ↑(a - b) x + ↑(b - c) xshow (a - b).1 x = a.1 x + (-1:ℝ)*b.1 x by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: ↑(H v e μ)
x: ↑Ω
a.h
↑(a - c) x = ↑(a - b) x + ↑(b - c) xrfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: ↑(H v e μ)
x: ↑Ω
a.h
↑(a - c) x = ↑a x + -1 * ↑b x + ↑(b - c) x
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: ↑(H v e μ)
a - c = a - b + (b - c)rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: ↑(H v e μ)
x: ↑Ω
a.h
↑(a - c) x = ↑a x + -1 * ↑b x + ↑(b - c) xshow (b - c).1 x = b.1 x + (-1:ℝ)*c.1 x by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: ↑(H v e μ)
x: ↑Ω
a.h
↑(a - c) x = ↑a x + -1 * ↑b x + ↑(b - c) xrfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: ↑(H v e μ)
x: ↑Ω
a.h
↑(a - c) x = ↑a x + -1 * ↑b x + (↑b x + -1 * ↑c x)
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: ↑(H v e μ)
a - c = a - b + (b - c)rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: ↑(H v e μ)
x: ↑Ω
a.h
↑(a - c) x = ↑a x + -1 * ↑b x + (↑b x + -1 * ↑c x)show a.1 x + -1 * b.1 x + (b.1 x + -1 * c.1 x) = a.1 x + (-1:ℝ)*c.1 x by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: ↑(H v e μ)
x: ↑Ω
a.h
↑(a - c) x = ↑a x + -1 * ↑b x + (↑b x + -1 * ↑c x)ring
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: ↑(H v e μ)
x: ↑Ω
a.h
↑(a - c) x = ↑a x + -1 * ↑c x
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: ↑(H v e μ)
a - c = a - b + (b - c)rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: ↑(H v e μ)
x: ↑Ω
a.h
↑(a - c) x = ↑a x + -1 * ↑c xshow (a - c).1 x = a.1 x + (-1:ℝ)*c.1 x by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: ↑(H v e μ)
x: ↑Ω
a.h
↑(a - c) x = ↑a x + -1 * ↑c xrfl
Goals accomplished! 🐙]
Goals accomplished! 🐙
  }
Goals accomplished! 🐙
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: ↑(H v e μ)
dist a c ≤ dist a b + dist b crw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: ↑(H v e μ)
split_a_sub_c: a - c = a - b + (b - c)
‖a - c‖ ≤ ‖a - b‖ + ‖b - c‖split_a_sub_c
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: ↑(H v e μ)
split_a_sub_c: a - c = a - b + (b - c)
‖a - b + (b - c)‖ ≤ ‖a - b‖ + ‖b - c‖]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: ↑(H v e μ)
split_a_sub_c: a - c = a - b + (b - c)
‖a - b + (b - c)‖ ≤ ‖a - b‖ + ‖b - c‖

  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: ↑(H v e μ)
dist a c ≤ dist a b + dist b cexact ineq_add_norm (a - b) (b - c)
Goals accomplished! 🐙

end Dist

/-
 - We instanciate the `NormedAddCommGroup` and `InnerProductSpace ℝ` typeclasses for H.
-/

variable {v : eigen} {e : ℕ → Ω → ℝ} {μ : Measure Ω} [IsFiniteMeasure μ]

lemma coe_mul_nat_fun (f : H v e μ) (n : ℕ) : (λ (n : ℕ) (f : H v e μ) ↦ (n : ℝ) * f) n f = (n : ℝ) * f := by
Goals accomplished! 🐙 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f: ↑(H v e μ)
n: ℕ
(fun n f => ↑n * f) n f = ↑n * frfl
Goals accomplished! 🐙

lemma coe_mul_int_fun (f : H v e μ) (z : ℤ) : (λ (z : ℤ) (f : H v e μ) ↦ (z : ℝ) * f) z f = (z : ℝ) * f := by
Goals accomplished! 🐙 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f: ↑(H v e μ)
z: ℤ
(fun z f => ↑z * f) z f = ↑z * frfl
Goals accomplished! 🐙

lemma cast_nat_succ (n : ℕ) : ↑(n + 1) = (n : ℝ) + 1 := by
Goals accomplished! 🐙 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
n: ℕ
↑(n + 1) = ↑n + 1simp
[Meta.Tactic.simp.rewrite] @Nat.cast_add:1000, ↑(n + 1) ==> ↑n + ↑1
[Meta.Tactic.simp.rewrite] @Nat.cast_one:1000, ↑1 ==> 1
[Meta.Tactic.simp.rewrite] @eq_self:1000, ↑n + 1 = ↑n + 1 ==> True
Goals accomplished! 🐙

lemma mul_succ_eq (f : H v e μ) (n : ℕ) : ((n : ℝ) + 1) * f = ((n : ℝ) * f + f) := by
Goals accomplished! 🐙
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f: ↑(H v e μ)
n: ℕ
(↑n + 1) * f = ↑n * f + fext x
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f: ↑(H v e μ)
n: ℕ
x: ↑Ω
a.h
↑((↑n + 1) * f) x = ↑(↑n * f + f) x
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f: ↑(H v e μ)
n: ℕ
(↑n + 1) * f = ↑n * f + flet n' := (n : ℝ)
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f: ↑(H v e μ)
n: ℕ
x: ↑Ω
n':= ↑n: ℝ
a.h
↑((↑n + 1) * f) x = ↑(↑n * f + f) x
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f: ↑(H v e μ)
n: ℕ
(↑n + 1) * f = ↑n * f + frw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f: ↑(H v e μ)
n: ℕ
x: ↑Ω
n':= ↑n: ℝ
a.h
↑((↑n + 1) * f) x = ↑(↑n * f + f) xshow (n' * f + f).1 x = (n' * f).1 x + f.1 x by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f: ↑(H v e μ)
n: ℕ
x: ↑Ω
n':= ↑n: ℝ
a.h
↑((↑n + 1) * f) x = ↑(↑n * f + f) xrfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f: ↑(H v e μ)
n: ℕ
x: ↑Ω
n':= ↑n: ℝ
a.h
↑((↑n + 1) * f) x = ↑(n' * f) x + ↑f x
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f: ↑(H v e μ)
n: ℕ
(↑n + 1) * f = ↑n * f + frw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f: ↑(H v e μ)
n: ℕ
x: ↑Ω
n':= ↑n: ℝ
a.h
↑((↑n + 1) * f) x = ↑(n' * f) x + ↑f xshow (n' * f).1 x = n' * f.1 x by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f: ↑(H v e μ)
n: ℕ
x: ↑Ω
n':= ↑n: ℝ
a.h
↑((↑n + 1) * f) x = ↑(n' * f) x + ↑f xrfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f: ↑(H v e μ)
n: ℕ
x: ↑Ω
n':= ↑n: ℝ
a.h
↑((↑n + 1) * f) x = n' * ↑f x + ↑f x
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f: ↑(H v e μ)
n: ℕ
(↑n + 1) * f = ↑n * f + frw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f: ↑(H v e μ)
n: ℕ
x: ↑Ω
n':= ↑n: ℝ
a.h
↑((↑n + 1) * f) x = n' * ↑f x + ↑f xshow ((n' + (1:ℝ)) * f).1 x = (n' + 1) * f.1 x by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f: ↑(H v e μ)
n: ℕ
x: ↑Ω
n':= ↑n: ℝ
a.h
↑((↑n + 1) * f) x = n' * ↑f x + ↑f xrfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f: ↑(H v e μ)
n: ℕ
x: ↑Ω
n':= ↑n: ℝ
a.h
(n' + 1) * ↑f x = n' * ↑f x + ↑f x
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f: ↑(H v e μ)
n: ℕ
(↑n + 1) * f = ↑n * f + fring
Goals accomplished! 🐙

noncomputable instance : NormedAddCommGroup (H v e μ) where
  dist := λ f g ↦ dist f g
  edist := λ f g ↦ ENNReal.ofReal (dist f g)
  norm := λ f ↦ norm f
  add := λ f g ↦ f + g
  add_assoc := Group.H_add_assoc
  zero_add := Group.H_zero_add
  add_zero := Group.H_add_zero
  nsmul := λ n f ↦ (n : ℝ) * f
  neg := λ f ↦ -f
  zsmul := λ z f ↦ (z : ℝ) * f
  add_left_neg := Group.H_add_left_neg
  add_comm := Group.H_add_comm
  dist_self := Dist.H_dist_self
  dist_comm := Dist.H_dist_comm
  dist_triangle := Dist.H_dist_triangle
  edist_dist := λ f g ↦ rfl
  eq_of_dist_eq_zero := Dist.H_eq_of_dist_eq_zero
  dist_eq := λ x y ↦ by
Goals accomplished! 🐙 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
x, y: ↑(H v e μ)
dist x y = ‖x - y‖rfl
Goals accomplished! 🐙
  nsmul_zero := by
Goals accomplished! 🐙 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (x : ↑(H v e μ)), (fun n f => ↑n * f) 0 x = 0{
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (x : ↑(H v e μ)), (fun n f => ↑n * f) 0 x = 0
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (x : ↑(H v e μ)), (fun n f => ↑n * f) 0 x = 0intro f
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f: ↑(H v e μ)
(fun n f => ↑n * f) 0 f = 0
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (x : ↑(H v e μ)), (fun n f => ↑n * f) 0 x = 0rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f: ↑(H v e μ)
(fun n f => ↑n * f) 0 f = 0coe_mul_nat_fun f 0
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f: ↑(H v e μ)
↑0 * f = 0]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f: ↑(H v e μ)
↑0 * f = 0
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (x : ↑(H v e μ)), (fun n f => ↑n * f) 0 x = 0rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f: ↑(H v e μ)
↑0 * f = 0show ((0 : ℕ) : ℝ) = (0 : ℝ) by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f: ↑(H v e μ)
↑0 * f = 0simp
[Meta.Tactic.simp.rewrite] CharP.cast_eq_zero:1000, ↑0 ==> 0
[Meta.Tactic.simp.rewrite] @eq_self:1000, 0 = 0 ==> True
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f: ↑(H v e μ)
0 * f = 0
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (x : ↑(H v e μ)), (fun n f => ↑n * f) 0 x = 0ext x
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f: ↑(H v e μ)
x: ↑Ω
a.h
↑(0 * f) x = ↑0 x
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (x : ↑(H v e μ)), (fun n f => ↑n * f) 0 x = 0rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f: ↑(H v e μ)
x: ↑Ω
a.h
↑(0 * f) x = ↑0 xshow ((0:ℝ) * f).1 x = (0:ℝ) * f.1 x by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f: ↑(H v e μ)
x: ↑Ω
a.h
↑(0 * f) x = ↑0 xrfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f: ↑(H v e μ)
x: ↑Ω
a.h
0 * ↑f x = ↑0 x
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (x : ↑(H v e μ)), (fun n f => ↑n * f) 0 x = 0rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f: ↑(H v e μ)
x: ↑Ω
a.h
0 * ↑f x = ↑0 xshow (0 : H v e μ).1 x = 0 by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f: ↑(H v e μ)
x: ↑Ω
a.h
0 * ↑f x = ↑0 xrfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f: ↑(H v e μ)
x: ↑Ω
a.h
0 * ↑f x = 0
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (x : ↑(H v e μ)), (fun n f => ↑n * f) 0 x = 0ring
Goals accomplished! 🐙
  }
Goals accomplished! 🐙
  nsmul_succ := by
Goals accomplished! 🐙 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (n : ℕ) (x : ↑(H v e μ)), (fun n f => ↑n * f) (n + 1) x = (fun n f => ↑n * f) n x + x{
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (n : ℕ) (x : ↑(H v e μ)), (fun n f => ↑n * f) (n + 1) x = (fun n f => ↑n * f) n x + x
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (n : ℕ) (x : ↑(H v e μ)), (fun n f => ↑n * f) (n + 1) x = (fun n f => ↑n * f) n x + xintro n f
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
n: ℕ
f: ↑(H v e μ)
(fun n f => ↑n * f) (n + 1) f = (fun n f => ↑n * f) n f + f
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (n : ℕ) (x : ↑(H v e μ)), (fun n f => ↑n * f) (n + 1) x = (fun n f => ↑n * f) n x + xrw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
n: ℕ
f: ↑(H v e μ)
(fun n f => ↑n * f) (n + 1) f = (fun n f => ↑n * f) n f + fcoe_mul_nat_fun f (n+1),
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
n: ℕ
f: ↑(H v e μ)
↑(n + 1) * f = (fun n f => ↑n * f) n f + f 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
n: ℕ
f: ↑(H v e μ)
(fun n f => ↑n * f) (n + 1) f = (fun n f => ↑n * f) n f + fcast_nat_succ n,
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
n: ℕ
f: ↑(H v e μ)
(↑n + 1) * f = (fun n f => ↑n * f) n f + f 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
n: ℕ
f: ↑(H v e μ)
(fun n f => ↑n * f) (n + 1) f = (fun n f => ↑n * f) n f + fcoe_mul_nat_fun f n
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
n: ℕ
f: ↑(H v e μ)
(↑n + 1) * f = ↑n * f + f]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
n: ℕ
f: ↑(H v e μ)
(↑n + 1) * f = ↑n * f + f
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (n : ℕ) (x : ↑(H v e μ)), (fun n f => ↑n * f) (n + 1) x = (fun n f => ↑n * f) n x + xexact mul_succ_eq f n
Goals accomplished! 🐙
  }
Goals accomplished! 🐙
  zsmul_zero' := by
Goals accomplished! 🐙 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (a : ↑(H v e μ)), (fun z f => ↑z * f) 0 a = 0{
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (a : ↑(H v e μ)), (fun z f => ↑z * f) 0 a = 0
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (a : ↑(H v e μ)), (fun z f => ↑z * f) 0 a = 0intro f
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f: ↑(H v e μ)
(fun z f => ↑z * f) 0 f = 0
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (a : ↑(H v e μ)), (fun z f => ↑z * f) 0 a = 0rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f: ↑(H v e μ)
(fun z f => ↑z * f) 0 f = 0coe_mul_int_fun f 0
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f: ↑(H v e μ)
↑0 * f = 0]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f: ↑(H v e μ)
↑0 * f = 0
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (a : ↑(H v e μ)), (fun z f => ↑z * f) 0 a = 0rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f: ↑(H v e μ)
↑0 * f = 0Int.cast_zero
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f: ↑(H v e μ)
0 * f = 0]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f: ↑(H v e μ)
0 * f = 0
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (a : ↑(H v e μ)), (fun z f => ↑z * f) 0 a = 0ext x
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f: ↑(H v e μ)
x: ↑Ω
a.h
↑(0 * f) x = ↑0 x
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (a : ↑(H v e μ)), (fun z f => ↑z * f) 0 a = 0rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f: ↑(H v e μ)
x: ↑Ω
a.h
↑(0 * f) x = ↑0 xshow ((0 : ℝ) * f).1 x = 0 * f.1 x by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f: ↑(H v e μ)
x: ↑Ω
a.h
↑(0 * f) x = ↑0 xrfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f: ↑(H v e μ)
x: ↑Ω
a.h
0 * ↑f x = ↑0 x
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (a : ↑(H v e μ)), (fun z f => ↑z * f) 0 a = 0rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f: ↑(H v e μ)
x: ↑Ω
a.h
0 * ↑f x = ↑0 xshow (0 : H v e μ).1 x = 0 by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f: ↑(H v e μ)
x: ↑Ω
a.h
0 * ↑f x = ↑0 xrfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f: ↑(H v e μ)
x: ↑Ω
a.h
0 * ↑f x = 0
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (a : ↑(H v e μ)), (fun z f => ↑z * f) 0 a = 0ring
Goals accomplished! 🐙
  }
Goals accomplished! 🐙
  zsmul_succ' := by
Goals accomplished! 🐙 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (n : ℕ) (a : ↑(H v e μ)), (fun z f => ↑z * f) (Int.ofNat (Nat.succ n)) a = (fun z f => ↑z * f) (Int.ofNat n) a + a{
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (n : ℕ) (a : ↑(H v e μ)), (fun z f => ↑z * f) (Int.ofNat (Nat.succ n)) a = (fun z f => ↑z * f) (Int.ofNat n) a + a
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (n : ℕ) (a : ↑(H v e μ)), (fun z f => ↑z * f) (Int.ofNat (Nat.succ n)) a = (fun z f => ↑z * f) (Int.ofNat n) a + aintro n f
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
n: ℕ
f: ↑(H v e μ)
(fun z f => ↑z * f) (Int.ofNat (Nat.succ n)) f = (fun z f => ↑z * f) (Int.ofNat n) f + f
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (n : ℕ) (a : ↑(H v e μ)), (fun z f => ↑z * f) (Int.ofNat (Nat.succ n)) a = (fun z f => ↑z * f) (Int.ofNat n) a + arw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
n: ℕ
f: ↑(H v e μ)
(fun z f => ↑z * f) (Int.ofNat (Nat.succ n)) f = (fun z f => ↑z * f) (Int.ofNat n) f + fshow Nat.succ n = n + 1 by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
n: ℕ
f: ↑(H v e μ)
(fun z f => ↑z * f) (Int.ofNat (Nat.succ n)) f = (fun z f => ↑z * f) (Int.ofNat n) f + frfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
n: ℕ
f: ↑(H v e μ)
(fun z f => ↑z * f) (Int.ofNat (n + 1)) f = (fun z f => ↑z * f) (Int.ofNat n) f + f
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (n : ℕ) (a : ↑(H v e μ)), (fun z f => ↑z * f) (Int.ofNat (Nat.succ n)) a = (fun z f => ↑z * f) (Int.ofNat n) a + arw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
n: ℕ
f: ↑(H v e μ)
(fun z f => ↑z * f) (Int.ofNat (n + 1)) f = (fun z f => ↑z * f) (Int.ofNat n) f + fcoe_mul_int_fun f (Int.ofNat (n + 1))
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
n: ℕ
f: ↑(H v e μ)
↑(Int.ofNat (n + 1)) * f = (fun z f => ↑z * f) (Int.ofNat n) f + f]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
n: ℕ
f: ↑(H v e μ)
↑(Int.ofNat (n + 1)) * f = (fun z f => ↑z * f) (Int.ofNat n) f + f
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (n : ℕ) (a : ↑(H v e μ)), (fun z f => ↑z * f) (Int.ofNat (Nat.succ n)) a = (fun z f => ↑z * f) (Int.ofNat n) a + arw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
n: ℕ
f: ↑(H v e μ)
↑(Int.ofNat (n + 1)) * f = (fun z f => ↑z * f) (Int.ofNat n) f + fshow (Int.ofNat (n + 1) : ℝ) = ↑(n + 1) by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
n: ℕ
f: ↑(H v e μ)
↑(Int.ofNat (n + 1)) * f = (fun z f => ↑z * f) (Int.ofNat n) f + frfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
n: ℕ
f: ↑(H v e μ)
↑(n + 1) * f = (fun z f => ↑z * f) (Int.ofNat n) f + f
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (n : ℕ) (a : ↑(H v e μ)), (fun z f => ↑z * f) (Int.ofNat (Nat.succ n)) a = (fun z f => ↑z * f) (Int.ofNat n) a + arw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
n: ℕ
f: ↑(H v e μ)
↑(n + 1) * f = (fun z f => ↑z * f) (Int.ofNat n) f + fcast_nat_succ n
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
n: ℕ
f: ↑(H v e μ)
(↑n + 1) * f = (fun z f => ↑z * f) (Int.ofNat n) f + f]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
n: ℕ
f: ↑(H v e μ)
(↑n + 1) * f = (fun z f => ↑z * f) (Int.ofNat n) f + f
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (n : ℕ) (a : ↑(H v e μ)), (fun z f => ↑z * f) (Int.ofNat (Nat.succ n)) a = (fun z f => ↑z * f) (Int.ofNat n) a + arw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
n: ℕ
f: ↑(H v e μ)
(↑n + 1) * f = (fun z f => ↑z * f) (Int.ofNat n) f + fcoe_mul_int_fun f (Int.ofNat n)
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
n: ℕ
f: ↑(H v e μ)
(↑n + 1) * f = ↑(Int.ofNat n) * f + f]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
n: ℕ
f: ↑(H v e μ)
(↑n + 1) * f = ↑(Int.ofNat n) * f + f
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (n : ℕ) (a : ↑(H v e μ)), (fun z f => ↑z * f) (Int.ofNat (Nat.succ n)) a = (fun z f => ↑z * f) (Int.ofNat n) a + arw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
n: ℕ
f: ↑(H v e μ)
(↑n + 1) * f = ↑(Int.ofNat n) * f + fshow (Int.ofNat n : ℝ) = (n : ℝ) by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
n: ℕ
f: ↑(H v e μ)
(↑n + 1) * f = ↑(Int.ofNat n) * f + frfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
n: ℕ
f: ↑(H v e μ)
(↑n + 1) * f = ↑n * f + f
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (n : ℕ) (a : ↑(H v e μ)), (fun z f => ↑z * f) (Int.ofNat (Nat.succ n)) a = (fun z f => ↑z * f) (Int.ofNat n) a + aexact mul_succ_eq f n
Goals accomplished! 🐙
  }
Goals accomplished! 🐙
  zsmul_neg' := by
Goals accomplished! 🐙 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (n : ℕ) (a : ↑(H v e μ)), (fun z f => ↑z * f) (Int.negSucc n) a = -(fun z f => ↑z * f) (↑(Nat.succ n)) a{
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (n : ℕ) (a : ↑(H v e μ)), (fun z f => ↑z * f) (Int.negSucc n) a = -(fun z f => ↑z * f) (↑(Nat.succ n)) a
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (n : ℕ) (a : ↑(H v e μ)), (fun z f => ↑z * f) (Int.negSucc n) a = -(fun z f => ↑z * f) (↑(Nat.succ n)) aintro n f
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
n: ℕ
f: ↑(H v e μ)
(fun z f => ↑z * f) (Int.negSucc n) f = -(fun z f => ↑z * f) (↑(Nat.succ n)) f
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (n : ℕ) (a : ↑(H v e μ)), (fun z f => ↑z * f) (Int.negSucc n) a = -(fun z f => ↑z * f) (↑(Nat.succ n)) arw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
n: ℕ
f: ↑(H v e μ)
(fun z f => ↑z * f) (Int.negSucc n) f = -(fun z f => ↑z * f) (↑(Nat.succ n)) fshow Int.negSucc n = -(n + 1) by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
n: ℕ
f: ↑(H v e μ)
(fun z f => ↑z * f) (Int.negSucc n) f = -(fun z f => ↑z * f) (↑(Nat.succ n)) frfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
n: ℕ
f: ↑(H v e μ)
(fun z f => ↑z * f) (-(↑n + 1)) f = -(fun z f => ↑z * f) (↑(Nat.succ n)) f
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (n : ℕ) (a : ↑(H v e μ)), (fun z f => ↑z * f) (Int.negSucc n) a = -(fun z f => ↑z * f) (↑(Nat.succ n)) arw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
n: ℕ
f: ↑(H v e μ)
(fun z f => ↑z * f) (-(↑n + 1)) f = -(fun z f => ↑z * f) (↑(Nat.succ n)) fshow Nat.succ n = n + 1 by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
n: ℕ
f: ↑(H v e μ)
(fun z f => ↑z * f) (-(↑n + 1)) f = -(fun z f => ↑z * f) (↑(Nat.succ n)) frfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
n: ℕ
f: ↑(H v e μ)
(fun z f => ↑z * f) (-(↑n + 1)) f = -(fun z f => ↑z * f) (↑(n + 1)) f
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (n : ℕ) (a : ↑(H v e μ)), (fun z f => ↑z * f) (Int.negSucc n) a = -(fun z f => ↑z * f) (↑(Nat.succ n)) arw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
n: ℕ
f: ↑(H v e μ)
(fun z f => ↑z * f) (-(↑n + 1)) f = -(fun z f => ↑z * f) (↑(n + 1)) fcoe_mul_int_fun f (-(n + 1))
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
n: ℕ
f: ↑(H v e μ)
↑(-(↑n + 1)) * f = -(fun z f => ↑z * f) (↑(n + 1)) f]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
n: ℕ
f: ↑(H v e μ)
↑(-(↑n + 1)) * f = -(fun z f => ↑z * f) (↑(n + 1)) f
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (n : ℕ) (a : ↑(H v e μ)), (fun z f => ↑z * f) (Int.negSucc n) a = -(fun z f => ↑z * f) (↑(Nat.succ n)) arw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
n: ℕ
f: ↑(H v e μ)
↑(-(↑n + 1)) * f = -(fun z f => ↑z * f) (↑(n + 1)) fshow -((fun (z:ℤ) (f:H v e μ) ↦ (z:ℝ) * f) (↑(n + 1)) f) = (-1 : ℝ) * ((fun (z:ℤ) (f:H v e μ) ↦ (z:ℝ) * f) (↑(n + 1)) f) by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
n: ℕ
f: ↑(H v e μ)
↑(-(↑n + 1)) * f = -(fun z f => ↑z * f) (↑(n + 1)) frfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
n: ℕ
f: ↑(H v e μ)
↑(-(↑n + 1)) * f = -1 * (fun z f => ↑z * f) (↑(n + 1)) f

    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (n : ℕ) (a : ↑(H v e μ)), (fun z f => ↑z * f) (Int.negSucc n) a = -(fun z f => ↑z * f) (↑(Nat.succ n)) arw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
n: ℕ
f: ↑(H v e μ)
↑(-(↑n + 1)) * f = -1 * (fun z f => ↑z * f) (↑(n + 1)) fshow (↑(n + 1) : ℤ) = (n : ℤ) + 1 by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
n: ℕ
f: ↑(H v e μ)
↑(-(↑n + 1)) * f = -1 * (fun z f => ↑z * f) (↑(n + 1)) frfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
n: ℕ
f: ↑(H v e μ)
↑(-(↑n + 1)) * f = -1 * (fun z f => ↑z * f) (↑n + 1) f
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (n : ℕ) (a : ↑(H v e μ)), (fun z f => ↑z * f) (Int.negSucc n) a = -(fun z f => ↑z * f) (↑(Nat.succ n)) arw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
n: ℕ
f: ↑(H v e μ)
↑(-(↑n + 1)) * f = -1 * (fun z f => ↑z * f) (↑n + 1) fcoe_mul_int_fun f (n + 1)
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
n: ℕ
f: ↑(H v e μ)
↑(-(↑n + 1)) * f = -1 * (↑(↑n + 1) * f)]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
n: ℕ
f: ↑(H v e μ)
↑(-(↑n + 1)) * f = -1 * (↑(↑n + 1) * f)
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (n : ℕ) (a : ↑(H v e μ)), (fun z f => ↑z * f) (Int.negSucc n) a = -(fun z f => ↑z * f) (↑(Nat.succ n)) arw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
n: ℕ
f: ↑(H v e μ)
↑(-(↑n + 1)) * f = -1 * (↑(↑n + 1) * f)show (↑((n : ℤ) + 1) : ℝ) = (n : ℝ) + 1 by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
n: ℕ
f: ↑(H v e μ)
↑(-(↑n + 1)) * f = -1 * (↑(↑n + 1) * f)simp
[Meta.Tactic.simp.rewrite] @Int.cast_add:1000, ↑(↑n + 1) ==> ↑↑n + ↑1
[Meta.Tactic.simp.rewrite] @Int.cast_natCast:10000, ↑↑n ==> ↑n
[Meta.Tactic.simp.rewrite] @Int.cast_one:1000, ↑1 ==> 1
[Meta.Tactic.simp.rewrite] @eq_self:1000, ↑n + 1 = ↑n + 1 ==> True
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
n: ℕ
f: ↑(H v e μ)
↑(-(↑n + 1)) * f = -1 * ((↑n + 1) * f)
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (n : ℕ) (a : ↑(H v e μ)), (fun z f => ↑z * f) (Int.negSucc n) a = -(fun z f => ↑z * f) (↑(Nat.succ n)) arw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
n: ℕ
f: ↑(H v e μ)
↑(-(↑n + 1)) * f = -1 * ((↑n + 1) * f)show (↑(- ((n : ℤ) + 1)) : ℝ) = -((n : ℝ) + 1) by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
n: ℕ
f: ↑(H v e μ)
↑(-(↑n + 1)) * f = -1 * ((↑n + 1) * f)simp
[Meta.Tactic.simp.rewrite] @neg_add_rev:1000, -(↑n + 1) ==> -1 + -↑n
[Meta.Tactic.simp.rewrite] @Int.cast_add:1000, ↑(-1 + -↑n) ==> ↑(-1) + ↑(-↑n)
[Meta.Tactic.simp.rewrite] @Int.cast_neg:1000, ↑(-1) ==> -↑1
[Meta.Tactic.simp.rewrite] @Int.cast_one:1000, ↑1 ==> 1
[Meta.Tactic.simp.rewrite] @Int.cast_neg:1000, ↑(-↑n) ==> -↑↑n
[Meta.Tactic.simp.rewrite] @Int.cast_natCast:10000, ↑↑n ==> ↑n
[Meta.Tactic.simp.rewrite] @neg_add_rev:1000, -(↑n + 1) ==> -1 + -↑n
[Meta.Tactic.simp.rewrite] @eq_self:1000, -1 + -↑n = -1 + -↑n ==> True
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
n: ℕ
f: ↑(H v e μ)
-(↑n + 1) * f = -1 * ((↑n + 1) * f)
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (n : ℕ) (a : ↑(H v e μ)), (fun z f => ↑z * f) (Int.negSucc n) a = -(fun z f => ↑z * f) (↑(Nat.succ n)) aext x
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
n: ℕ
f: ↑(H v e μ)
x: ↑Ω
a.h
↑(-(↑n + 1) * f) x = ↑(-1 * ((↑n + 1) * f)) x
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (n : ℕ) (a : ↑(H v e μ)), (fun z f => ↑z * f) (Int.negSucc n) a = -(fun z f => ↑z * f) (↑(Nat.succ n)) arw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
n: ℕ
f: ↑(H v e μ)
x: ↑Ω
a.h
↑(-(↑n + 1) * f) x = ↑(-1 * ((↑n + 1) * f)) xshow (-((n : ℝ) + (1 : ℝ)) * f).1 x = -((n : ℝ) + 1) * f.1 x by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
n: ℕ
f: ↑(H v e μ)
x: ↑Ω
a.h
↑(-(↑n + 1) * f) x = ↑(-1 * ((↑n + 1) * f)) xrfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
n: ℕ
f: ↑(H v e μ)
x: ↑Ω
a.h
-(↑n + 1) * ↑f x = ↑(-1 * ((↑n + 1) * f)) x
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (n : ℕ) (a : ↑(H v e μ)), (fun z f => ↑z * f) (Int.negSucc n) a = -(fun z f => ↑z * f) (↑(Nat.succ n)) arw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
n: ℕ
f: ↑(H v e μ)
x: ↑Ω
a.h
-(↑n + 1) * ↑f x = ↑(-1 * ((↑n + 1) * f)) xshow ( (-1 : ℝ) * ( ((n : ℝ) + (1 : ℝ)) * f ) ).1 x = (-1 : ℝ) * (((n : ℝ) + (1 : ℝ)) * f).1 x by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
n: ℕ
f: ↑(H v e μ)
x: ↑Ω
a.h
-(↑n + 1) * ↑f x = ↑(-1 * ((↑n + 1) * f)) xrfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
n: ℕ
f: ↑(H v e μ)
x: ↑Ω
a.h
-(↑n + 1) * ↑f x = -1 * ↑((↑n + 1) * f) x
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (n : ℕ) (a : ↑(H v e μ)), (fun z f => ↑z * f) (Int.negSucc n) a = -(fun z f => ↑z * f) (↑(Nat.succ n)) arw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
n: ℕ
f: ↑(H v e μ)
x: ↑Ω
a.h
-(↑n + 1) * ↑f x = -1 * ↑((↑n + 1) * f) xshow (((n : ℝ) + (1 : ℝ)) * f).1 x = ((n : ℝ) + (1 : ℝ)) * f.1 x by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
n: ℕ
f: ↑(H v e μ)
x: ↑Ω
a.h
-(↑n + 1) * ↑f x = -1 * ↑((↑n + 1) * f) xrfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
n: ℕ
f: ↑(H v e μ)
x: ↑Ω
a.h
-(↑n + 1) * ↑f x = -1 * ((↑n + 1) * ↑f x)
    
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (n : ℕ) (a : ↑(H v e μ)), (fun z f => ↑z * f) (Int.negSucc n) a = -(fun z f => ↑z * f) (↑(Nat.succ n)) aring
Goals accomplished! 🐙
  }
Goals accomplished! 🐙

open Inner

noncomputable instance : InnerProductSpace ℝ (H v e μ) where
smul := λ a f ↦ a * f
one_smul := by
Goals accomplished! 🐙 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (b : ↑(H v e μ)), 1 • b = b{
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (b : ↑(H v e μ)), 1 • b = b
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (b : ↑(H v e μ)), 1 • b = bintro b
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
b: ↑(H v e μ)
1 • b = b
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (b : ↑(H v e μ)), 1 • b = bext x
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
b: ↑(H v e μ)
x: ↑Ω
a.h
↑(1 • b) x = ↑b x
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (b : ↑(H v e μ)), 1 • b = brw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
b: ↑(H v e μ)
x: ↑Ω
a.h
↑(1 • b) x = ↑b xshow ((1:ℝ) • b).1 x = (1:ℝ) * b.1 x by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
b: ↑(H v e μ)
x: ↑Ω
a.h
↑(1 • b) x = ↑b xrfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
b: ↑(H v e μ)
x: ↑Ω
a.h
1 * ↑b x = ↑b x
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (b : ↑(H v e μ)), 1 • b = bring
Goals accomplished! 🐙
}
Goals accomplished! 🐙
mul_smul := by
Goals accomplished! 🐙 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (x y : ℝ) (b : ↑(H v e μ)), (x * y) • b = x • y • b{
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (x y : ℝ) (b : ↑(H v e μ)), (x * y) • b = x • y • b
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (x y : ℝ) (b : ↑(H v e μ)), (x * y) • b = x • y • bintro x y b
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
x, y: ℝ
b: ↑(H v e μ)
(x * y) • b = x • y • b
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (x y : ℝ) (b : ↑(H v e μ)), (x * y) • b = x • y • bext e
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e✝: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
x, y: ℝ
b: ↑(H v e✝ μ)
e: ↑Ω
a.h
↑((x * y) • b) e = ↑(x • y • b) e
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (x y : ℝ) (b : ↑(H v e μ)), (x * y) • b = x • y • brw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e✝: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
x, y: ℝ
b: ↑(H v e✝ μ)
e: ↑Ω
a.h
↑((x * y) • b) e = ↑(x • y • b) eshow ((x*y) • b).1 e = (x*y) * b.1 e by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e✝: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
x, y: ℝ
b: ↑(H v e✝ μ)
e: ↑Ω
a.h
↑((x * y) • b) e = ↑(x • y • b) erfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e✝: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
x, y: ℝ
b: ↑(H v e✝ μ)
e: ↑Ω
a.h
x * y * ↑b e = ↑(x • y • b) e
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (x y : ℝ) (b : ↑(H v e μ)), (x * y) • b = x • y • brw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e✝: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
x, y: ℝ
b: ↑(H v e✝ μ)
e: ↑Ω
a.h
x * y * ↑b e = ↑(x • y • b) eshow (x • y • b).1 e = x * y • b.1 e by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e✝: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
x, y: ℝ
b: ↑(H v e✝ μ)
e: ↑Ω
a.h
x * y * ↑b e = ↑(x • y • b) erfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e✝: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
x, y: ℝ
b: ↑(H v e✝ μ)
e: ↑Ω
a.h
x * y * ↑b e = x * y • ↑b e
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (x y : ℝ) (b : ↑(H v e μ)), (x * y) • b = x • y • brw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e✝: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
x, y: ℝ
b: ↑(H v e✝ μ)
e: ↑Ω
a.h
x * y * ↑b e = x * y • ↑b eshow y • b.1 e = y * b.1 e by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e✝: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
x, y: ℝ
b: ↑(H v e✝ μ)
e: ↑Ω
a.h
x * y * ↑b e = x * y • ↑b erfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e✝: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
x, y: ℝ
b: ↑(H v e✝ μ)
e: ↑Ω
a.h
x * y * ↑b e = x * (y * ↑b e)
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (x y : ℝ) (b : ↑(H v e μ)), (x * y) • b = x • y • bring
Goals accomplished! 🐙
}
Goals accomplished! 🐙
smul_zero := by
Goals accomplished! 🐙 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (a : ℝ), a • 0 = 0{
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (a : ℝ), a • 0 = 0
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (a : ℝ), a • 0 = 0intro a
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
a: ℝ
a • 0 = 0
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (a : ℝ), a • 0 = 0ext x
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
a: ℝ
x: ↑Ω
a.h
↑(a • 0) x = ↑0 x
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (a : ℝ), a • 0 = 0rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
a: ℝ
x: ↑Ω
a.h
↑(a • 0) x = ↑0 xshow (a • (0 : H v e μ)).1 x = a * 0 by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
a: ℝ
x: ↑Ω
a.h
↑(a • 0) x = ↑0 xrfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
a: ℝ
x: ↑Ω
a.h
a * 0 = ↑0 x
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (a : ℝ), a • 0 = 0rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
a: ℝ
x: ↑Ω
a.h
a * 0 = ↑0 xshow (0 : H v e μ).1 x = 0 by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
a: ℝ
x: ↑Ω
a.h
a * 0 = ↑0 xrfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
a: ℝ
x: ↑Ω
a.h
a * 0 = 0
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (a : ℝ), a • 0 = 0ring
Goals accomplished! 🐙
}
Goals accomplished! 🐙
smul_add := by
Goals accomplished! 🐙 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (a : ℝ) (x y : ↑(H v e μ)), a • (x + y) = a • x + a • y{
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (a : ℝ) (x y : ↑(H v e μ)), a • (x + y) = a • x + a • y
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (a : ℝ) (x y : ↑(H v e μ)), a • (x + y) = a • x + a • yintro a f g
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
a: ℝ
f, g: ↑(H v e μ)
a • (f + g) = a • f + a • g
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (a : ℝ) (x y : ↑(H v e μ)), a • (x + y) = a • x + a • yext x
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
a: ℝ
f, g: ↑(H v e μ)
x: ↑Ω
a.h
↑(a • (f + g)) x = ↑(a • f + a • g) x
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (a : ℝ) (x y : ↑(H v e μ)), a • (x + y) = a • x + a • yrw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
a: ℝ
f, g: ↑(H v e μ)
x: ↑Ω
a.h
↑(a • (f + g)) x = ↑(a • f + a • g) xshow (a • (f + g)).1 x = a * (f + g).1 x by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
a: ℝ
f, g: ↑(H v e μ)
x: ↑Ω
a.h
↑(a • (f + g)) x = ↑(a • f + a • g) xrfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
a: ℝ
f, g: ↑(H v e μ)
x: ↑Ω
a.h
a * ↑(f + g) x = ↑(a • f + a • g) x
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (a : ℝ) (x y : ↑(H v e μ)), a • (x + y) = a • x + a • yrw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
a: ℝ
f, g: ↑(H v e μ)
x: ↑Ω
a.h
a * ↑(f + g) x = ↑(a • f + a • g) xshow (f + g).1 x = f.1 x + g.1 x by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
a: ℝ
f, g: ↑(H v e μ)
x: ↑Ω
a.h
a * ↑(f + g) x = ↑(a • f + a • g) xrfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
a: ℝ
f, g: ↑(H v e μ)
x: ↑Ω
a.h
a * (↑f x + ↑g x) = ↑(a • f + a • g) x
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (a : ℝ) (x y : ↑(H v e μ)), a • (x + y) = a • x + a • yrw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
a: ℝ
f, g: ↑(H v e μ)
x: ↑Ω
a.h
a * (↑f x + ↑g x) = ↑(a • f + a • g) xshow (a • f + a • g).1 x = a * f.1 x + a * g.1 x by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
a: ℝ
f, g: ↑(H v e μ)
x: ↑Ω
a.h
a * (↑f x + ↑g x) = ↑(a • f + a • g) xrfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
a: ℝ
f, g: ↑(H v e μ)
x: ↑Ω
a.h
a * (↑f x + ↑g x) = a * ↑f x + a * ↑g x
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (a : ℝ) (x y : ↑(H v e μ)), a • (x + y) = a • x + a • yring
Goals accomplished! 🐙
}
Goals accomplished! 🐙
add_smul := by
Goals accomplished! 🐙 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (r s : ℝ) (x : ↑(H v e μ)), (r + s) • x = r • x + s • x{
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (r s : ℝ) (x : ↑(H v e μ)), (r + s) • x = r • x + s • x
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (r s : ℝ) (x : ↑(H v e μ)), (r + s) • x = r • x + s • xintro r s f
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
r, s: ℝ
f: ↑(H v e μ)
(r + s) • f = r • f + s • f
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (r s : ℝ) (x : ↑(H v e μ)), (r + s) • x = r • x + s • xext x
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
r, s: ℝ
f: ↑(H v e μ)
x: ↑Ω
a.h
↑((r + s) • f) x = ↑(r • f + s • f) x
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (r s : ℝ) (x : ↑(H v e μ)), (r + s) • x = r • x + s • xrw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
r, s: ℝ
f: ↑(H v e μ)
x: ↑Ω
a.h
↑((r + s) • f) x = ↑(r • f + s • f) xshow ((r + s) • f).1 x = (r + s) * f.1 x by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
r, s: ℝ
f: ↑(H v e μ)
x: ↑Ω
a.h
↑((r + s) • f) x = ↑(r • f + s • f) xrfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
r, s: ℝ
f: ↑(H v e μ)
x: ↑Ω
a.h
(r + s) * ↑f x = ↑(r • f + s • f) x
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (r s : ℝ) (x : ↑(H v e μ)), (r + s) • x = r • x + s • xrw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
r, s: ℝ
f: ↑(H v e μ)
x: ↑Ω
a.h
(r + s) * ↑f x = ↑(r • f + s • f) xshow (r • f + s • f).1 x = r * f.1 x + s * f.1 x by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
r, s: ℝ
f: ↑(H v e μ)
x: ↑Ω
a.h
(r + s) * ↑f x = ↑(r • f + s • f) xrfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
r, s: ℝ
f: ↑(H v e μ)
x: ↑Ω
a.h
(r + s) * ↑f x = r * ↑f x + s * ↑f x
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (r s : ℝ) (x : ↑(H v e μ)), (r + s) • x = r • x + s • xring
Goals accomplished! 🐙
}
Goals accomplished! 🐙
zero_smul := by
Goals accomplished! 🐙 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (x : ↑(H v e μ)), 0 • x = 0{
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (x : ↑(H v e μ)), 0 • x = 0
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (x : ↑(H v e μ)), 0 • x = 0intro f
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f: ↑(H v e μ)
0 • f = 0
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (x : ↑(H v e μ)), 0 • x = 0ext x
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f: ↑(H v e μ)
x: ↑Ω
a.h
↑(0 • f) x = ↑0 x
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (x : ↑(H v e μ)), 0 • x = 0rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f: ↑(H v e μ)
x: ↑Ω
a.h
↑(0 • f) x = ↑0 xshow ((0:ℝ) • f).1 x = (0:ℝ) * f.1 x by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f: ↑(H v e μ)
x: ↑Ω
a.h
↑(0 • f) x = ↑0 xrfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f: ↑(H v e μ)
x: ↑Ω
a.h
0 * ↑f x = ↑0 x
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (x : ↑(H v e μ)), 0 • x = 0rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f: ↑(H v e μ)
x: ↑Ω
a.h
0 * ↑f x = ↑0 xshow (0 : H v e μ).1 x = 0 by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f: ↑(H v e μ)
x: ↑Ω
a.h
0 * ↑f x = ↑0 xrfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f: ↑(H v e μ)
x: ↑Ω
a.h
0 * ↑f x = 0
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (x : ↑(H v e μ)), 0 • x = 0ring
Goals accomplished! 🐙
}
Goals accomplished! 🐙
norm_smul_le := by
Goals accomplished! 🐙 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (a : ℝ) (b : ↑(H v e μ)), ‖a • b‖ ≤ ‖a‖ * ‖b‖{
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (a : ℝ) (b : ↑(H v e μ)), ‖a • b‖ ≤ ‖a‖ * ‖b‖
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (a : ℝ) (b : ↑(H v e μ)), ‖a • b‖ ≤ ‖a‖ * ‖b‖intro r f
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
r: ℝ
f: ↑(H v e μ)
‖r • f‖ ≤ ‖r‖ * ‖f‖
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (a : ℝ) (b : ↑(H v e μ)), ‖a • b‖ ≤ ‖a‖ * ‖b‖apply le_of_eq
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
r: ℝ
f: ↑(H v e μ)
a
‖r • f‖ = ‖r‖ * ‖f‖
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (a : ℝ) (b : ↑(H v e μ)), ‖a • b‖ ≤ ‖a‖ * ‖b‖rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
r: ℝ
f: ↑(H v e μ)
a
‖r • f‖ = ‖r‖ * ‖f‖show r • f = r * f by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
r: ℝ
f: ↑(H v e μ)
a
‖r • f‖ = ‖r‖ * ‖f‖rfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
r: ℝ
f: ↑(H v e μ)
a
‖r * f‖ = ‖r‖ * ‖f‖
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (a : ℝ) (b : ↑(H v e μ)), ‖a • b‖ ≤ ‖a‖ * ‖b‖rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
r: ℝ
f: ↑(H v e μ)
a
‖r * f‖ = ‖r‖ * ‖f‖show ‖r‖ = |r| by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
r: ℝ
f: ↑(H v e μ)
a
‖r * f‖ = ‖r‖ * ‖f‖rfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
r: ℝ
f: ↑(H v e μ)
a
‖r * f‖ = |r| * ‖f‖
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (a : ℝ) (b : ↑(H v e μ)), ‖a • b‖ ≤ ‖a‖ * ‖b‖rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
r: ℝ
f: ↑(H v e μ)
a
‖r * f‖ = |r| * ‖f‖←mul_self_inj (H_norm_nonneg (r * f)) (Left.mul_nonneg (abs_nonneg r) (H_norm_nonneg (f)))
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
r: ℝ
f: ↑(H v e μ)
a
‖r * f‖ * ‖r * f‖ = |r| * ‖f‖ * (|r| * ‖f‖)]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
r: ℝ
f: ↑(H v e μ)
a
‖r * f‖ * ‖r * f‖ = |r| * ‖f‖ * (|r| * ‖f‖)
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (a : ℝ) (b : ↑(H v e μ)), ‖a • b‖ ≤ ‖a‖ * ‖b‖rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
r: ℝ
f: ↑(H v e μ)
a
‖r * f‖ * ‖r * f‖ = |r| * ‖f‖ * (|r| * ‖f‖)show ‖r * f‖ * ‖r * f‖ = ‖r * f‖^2 by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
r: ℝ
f: ↑(H v e μ)
a
‖r * f‖ * ‖r * f‖ = |r| * ‖f‖ * (|r| * ‖f‖)ring
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
r: ℝ
f: ↑(H v e μ)
a
‖r * f‖ ^ 2 = |r| * ‖f‖ * (|r| * ‖f‖)
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (a : ℝ) (b : ↑(H v e μ)), ‖a • b‖ ≤ ‖a‖ * ‖b‖rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
r: ℝ
f: ↑(H v e μ)
a
‖r * f‖ ^ 2 = |r| * ‖f‖ * (|r| * ‖f‖)show |r| * ‖f‖ * (|r| * ‖f‖) = (|r| * ‖f‖)^2 by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
r: ℝ
f: ↑(H v e μ)
a
‖r * f‖ ^ 2 = |r| * ‖f‖ * (|r| * ‖f‖)ring
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
r: ℝ
f: ↑(H v e μ)
a
‖r * f‖ ^ 2 = (|r| * ‖f‖) ^ 2
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (a : ℝ) (b : ↑(H v e μ)), ‖a • b‖ ≤ ‖a‖ * ‖b‖rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
r: ℝ
f: ↑(H v e μ)
a
‖r * f‖ ^ 2 = (|r| * ‖f‖) ^ 2←inner_eq_sq_norm (r * f),
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
r: ℝ
f: ↑(H v e μ)
a
⟪r * f, r * f⟫_ℝ = (|r| * ‖f‖) ^ 2 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
r: ℝ
f: ↑(H v e μ)
a
‖r * f‖ ^ 2 = (|r| * ‖f‖) ^ 2inner_mul_left f (r * f) r
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
r: ℝ
f: ↑(H v e μ)
a
r * ⟪f, r * f⟫_ℝ = (|r| * ‖f‖) ^ 2]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
r: ℝ
f: ↑(H v e μ)
a
r * ⟪f, r * f⟫_ℝ = (|r| * ‖f‖) ^ 2
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (a : ℝ) (b : ↑(H v e μ)), ‖a • b‖ ≤ ‖a‖ * ‖b‖rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
r: ℝ
f: ↑(H v e μ)
a
r * ⟪f, r * f⟫_ℝ = (|r| * ‖f‖) ^ 2inner_symmetric,
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
r: ℝ
f: ↑(H v e μ)
a
r * ⟪r * f, f⟫_ℝ = (|r| * ‖f‖) ^ 2 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
r: ℝ
f: ↑(H v e μ)
a
r * ⟪f, r * f⟫_ℝ = (|r| * ‖f‖) ^ 2inner_mul_left f f r,
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
r: ℝ
f: ↑(H v e μ)
a
r * (r * ⟪f, f⟫_ℝ) = (|r| * ‖f‖) ^ 2 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
r: ℝ
f: ↑(H v e μ)
a
r * ⟪f, r * f⟫_ℝ = (|r| * ‖f‖) ^ 2inner_eq_sq_norm
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
r: ℝ
f: ↑(H v e μ)
a
r * (r * ‖f‖ ^ 2) = (|r| * ‖f‖) ^ 2]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
r: ℝ
f: ↑(H v e μ)
a
r * (r * ‖f‖ ^ 2) = (|r| * ‖f‖) ^ 2
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (a : ℝ) (b : ↑(H v e μ)), ‖a • b‖ ≤ ‖a‖ * ‖b‖rw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
r: ℝ
f: ↑(H v e μ)
a
r * (r * ‖f‖ ^ 2) = (|r| * ‖f‖) ^ 2show r * (r * ‖f‖^2) = r^2 * ‖f‖^2 by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
r: ℝ
f: ↑(H v e μ)
a
r * (r * ‖f‖ ^ 2) = (|r| * ‖f‖) ^ 2ring
Goals accomplished! 🐙,
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
r: ℝ
f: ↑(H v e μ)
a
r ^ 2 * ‖f‖ ^ 2 = (|r| * ‖f‖) ^ 2 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
r: ℝ
f: ↑(H v e μ)
a
r * (r * ‖f‖ ^ 2) = (|r| * ‖f‖) ^ 2←sq_abs r
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
r: ℝ
f: ↑(H v e μ)
a
|r| ^ 2 * ‖f‖ ^ 2 = (|r| * ‖f‖) ^ 2]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
r: ℝ
f: ↑(H v e μ)
a
|r| ^ 2 * ‖f‖ ^ 2 = (|r| * ‖f‖) ^ 2
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (a : ℝ) (b : ↑(H v e μ)), ‖a • b‖ ≤ ‖a‖ * ‖b‖ring
Goals accomplished! 🐙
}
Goals accomplished! 🐙
inner := H_inner
norm_sq_eq_inner := by
Goals accomplished! 🐙 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (x : ↑(H v e μ)), ‖x‖ ^ 2 = RCLike.re ⟪x, x⟫_ℝ{
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (x : ↑(H v e μ)), ‖x‖ ^ 2 = RCLike.re ⟪x, x⟫_ℝ
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (x : ↑(H v e μ)), ‖x‖ ^ 2 = RCLike.re ⟪x, x⟫_ℝintro f
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f: ↑(H v e μ)
‖f‖ ^ 2 = RCLike.re ⟪f, f⟫_ℝ
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (x : ↑(H v e μ)), ‖x‖ ^ 2 = RCLike.re ⟪x, x⟫_ℝrw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f: ↑(H v e μ)
‖f‖ ^ 2 = RCLike.re ⟪f, f⟫_ℝ←inner_eq_sq_norm
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f: ↑(H v e μ)
⟪f, f⟫_ℝ = RCLike.re ⟪f, f⟫_ℝ]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f: ↑(H v e μ)
⟪f, f⟫_ℝ = RCLike.re ⟪f, f⟫_ℝ
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (x : ↑(H v e μ)), ‖x‖ ^ 2 = RCLike.re ⟪x, x⟫_ℝrfl
Goals accomplished! 🐙
}
Goals accomplished! 🐙
conj_symm := by
Goals accomplished! 🐙 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (x y : ↑(H v e μ)), (starRingEnd ℝ) ⟪y, x⟫_ℝ = ⟪x, y⟫_ℝ{
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (x y : ↑(H v e μ)), (starRingEnd ℝ) ⟪y, x⟫_ℝ = ⟪x, y⟫_ℝ
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (x y : ↑(H v e μ)), (starRingEnd ℝ) ⟪y, x⟫_ℝ = ⟪x, y⟫_ℝintro f g
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
(starRingEnd ℝ) ⟪g, f⟫_ℝ = ⟪f, g⟫_ℝ
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (x y : ↑(H v e μ)), (starRingEnd ℝ) ⟪y, x⟫_ℝ = ⟪x, y⟫_ℝrw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
(starRingEnd ℝ) ⟪g, f⟫_ℝ = ⟪f, g⟫_ℝinner_symmetric f g
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
(starRingEnd ℝ) ⟪g, f⟫_ℝ = ⟪g, f⟫_ℝ]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
(starRingEnd ℝ) ⟪g, f⟫_ℝ = ⟪g, f⟫_ℝ
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (x y : ↑(H v e μ)), (starRingEnd ℝ) ⟪y, x⟫_ℝ = ⟪x, y⟫_ℝrfl
Goals accomplished! 🐙
}
Goals accomplished! 🐙
add_left := by
Goals accomplished! 🐙 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (x y z : ↑(H v e μ)), ⟪x + y, z⟫_ℝ = ⟪x, z⟫_ℝ + ⟪y, z⟫_ℝ{
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (x y z : ↑(H v e μ)), ⟪x + y, z⟫_ℝ = ⟪x, z⟫_ℝ + ⟪y, z⟫_ℝ
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (x y z : ↑(H v e μ)), ⟪x + y, z⟫_ℝ = ⟪x, z⟫_ℝ + ⟪y, z⟫_ℝintro f g h
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g, h: ↑(H v e μ)
⟪f + g, h⟫_ℝ = ⟪f, h⟫_ℝ + ⟪g, h⟫_ℝ
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (x y z : ↑(H v e μ)), ⟪x + y, z⟫_ℝ = ⟪x, z⟫_ℝ + ⟪y, z⟫_ℝexact H_inner_add_left f g h
Goals accomplished! 🐙
}
Goals accomplished! 🐙
smul_left := by
Goals accomplished! 🐙 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (x y : ↑(H v e μ)) (r : ℝ), ⟪r • x, y⟫_ℝ = (starRingEnd ℝ) r * ⟪x, y⟫_ℝ{
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (x y : ↑(H v e μ)) (r : ℝ), ⟪r • x, y⟫_ℝ = (starRingEnd ℝ) r * ⟪x, y⟫_ℝ
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (x y : ↑(H v e μ)) (r : ℝ), ⟪r • x, y⟫_ℝ = (starRingEnd ℝ) r * ⟪x, y⟫_ℝintro f g r
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
r: ℝ
⟪r • f, g⟫_ℝ = (starRingEnd ℝ) r * ⟪f, g⟫_ℝ
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (x y : ↑(H v e μ)) (r : ℝ), ⟪r • x, y⟫_ℝ = (starRingEnd ℝ) r * ⟪x, y⟫_ℝrw [
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
r: ℝ
⟪r • f, g⟫_ℝ = (starRingEnd ℝ) r * ⟪f, g⟫_ℝshow r • f = r * f by 
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
r: ℝ
⟪r • f, g⟫_ℝ = (starRingEnd ℝ) r * ⟪f, g⟫_ℝrfl
Goals accomplished! 🐙]
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
f, g: ↑(H v e μ)
r: ℝ
⟪r * f, g⟫_ℝ = (starRingEnd ℝ) r * ⟪f, g⟫_ℝ
  
d: ℕ
Ω: Set (Vector ℝ d)
inst✝¹: MeasureSpace ↑Ω
v: eigen
e: ℕ → ↑Ω → ℝ
μ: Measure ↑Ω
inst✝: IsFiniteMeasure μ
∀ (x y : ↑(H v e μ)) (r : ℝ), ⟪r • x, y⟫_ℝ = (starRingEnd ℝ) r * ⟪x, y⟫_ℝexact inner_mul_left f g r
Goals accomplished! 🐙
}
Goals accomplished! 🐙